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4.6: Limits at Infinity and Asymptotes
[ "article:topic", "horizontal asymptote", "end behavior", "authorname:openstax", "infinite limit at infinity", "limit at infinity", "oblique asymptote", "calcplot:yes", "license:ccbyncsa", "showtoc:no", "transcluded:yes" ]
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MTH 210 Calculus I
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Limits at Infinity and Horizontal Asymptotes
Infinite Limits at Infinity
Formal Definitions
End Behavior
End Behavior for Polynomial Functions
End Behavior for Algebraic Functions
Determining End Behavior for Transcendental Functions
Guidelines for Drawing the Graph of a Function
We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function \(f\) defined on an unbounded domain, we also need to know the behavior of \(f\) as \(x→±∞\). In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function \(f\).
We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity. Back in Introduction to Functions and Graphs, we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.
Recall that \(\lim_{x→a}f(x)=L\) means \(f(x)\) becomes arbitrarily close to \(L\) as long as \(x\) is sufficiently close to \(a\). We can extend this idea to limits at infinity. For example, consider the function \(f(x)=2+\frac{1}{x}\). As can be seen graphically in Figure and numerically in Table, as the values of \(x\) get larger, the values of \(f(x)\) approach \(2\). We say the limit as \(x\) approaches \(∞\) of \(f(x)\) is \(2\) and write \(\lim_{x→∞}f(x)=2\). Similarly, for \(x<0\), as the values \(|x|\) get larger, the values of \(f(x)\) approaches \(2\). We say the limit as \(x\) approaches \(−∞\) of \(f(x)\) is \(2\) and write \(\lim_{x→a}f(x)=2\).
Figure \(\PageIndex{1}\):The function approaches the asymptote \(y=2\) as \(x\) approaches \(±∞\).
Table \(\PageIndex{1}\): Values of a function \(f\) as \(x→±∞\)
\(x\)
10 100 1,000 10,000
\(2+\frac{1}{x}\)
2.1 2.01 2.001 2.0001
−10 −100 −1000 −10,000
More generally, for any function \(f\), we say the limit as \(x→∞\) of \(f(x)\) is \(L\) if \(f(x)\) becomes arbitrarily close to \(L\) as long as \(x\) is sufficiently large. In that case, we write \(\lim_{x→a}f(x)=L\). Similarly, we say the limit as \(x→−∞\) of \(f(x)\) is \(L\) if \(f(x)\) becomes arbitrarily close to \(L\) as long as \(x<0\) and \(|x|\) is sufficiently large. In that case, we write \(\lim_{x→−∞}f(x)=L\). We now look at the definition of a function having a limit at infinity.
Definition: limit at infinity (Informal)
If the values of \(f(x)\) become arbitrarily close to \(L\) as \(x\) becomes sufficiently large, we say the function \(f\) has a limit at infinity and write
\[\lim_{x→∞}f(x)=L.\]
If the values of \(f(x)\) becomes arbitrarily close to \(L\) for \(x<0\) as \(|x|\) becomes sufficiently large, we say that the function \(f\) has a limit at negative infinity and write
If the values \(f(x)\) are getting arbitrarily close to some finite value \(L\) as \(x→∞\) or \(x→−∞\), the graph of \(f\) approaches the line \(y=L\). In that case, the line \(y=L\) is a horizontal asymptote of \(f\) (Figure). For example, for the function \(f(x)=\frac{1}{x}\), since \(\lim_{x→∞}f(x)=0\), the line \(y=0\) is a horizontal asymptote of \(f(x)=\frac{1}{x}\).
Definition: horizontal asymptote
If \(\lim_{x→∞}f(x)=L\) or \(\lim_{x→−∞}f(x)=L\), we say the line \(y=L\) is a horizontal asymptote of \(f\).
Figure \(\PageIndex{2}\): (a) As \(x→∞\), the values of \(f\) are getting arbitrarily close to \(L\). The line \(y=L\) is a horizontal asymptote of \(f\). (b) As \(x→−∞\), the values of \(f\) are getting arbitrarily close to \(M\). The line \(y=M\) is a horizontal asymptote of \(f\).
A function cannot cross a vertical asymptote because the graph must approach infinity (or \( −∞\)) from at least one direction as \(x\) approaches the vertical asymptote. However, a function may cross a horizontal asymptote. In fact, a function may cross a horizontal asymptote an unlimited number of times. For example, the function \(f(x)=\frac{(cosx)}{x}+1\) shown in Figure intersects the horizontal asymptote \(y=1\) an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.
Figure \(\PageIndex{3}\): The graph of \(f(x)=(cosx)/x+1\) crosses its horizontal asymptote \(y=1\) an infinite number of times.
The algebraic limit laws and squeeze theorem we introduced in Introduction to Limits also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.
Example \(\PageIndex{1}\): Computing Limits at Infinity
For each of the following functions \(f\), valuate \(\lim_{x→∞}f(x)\) and \(\lim_{x→−∞}f(x)\). Determine the horizontal asymptote(s) for \(f\).
\(f(x)=5−\frac{2}{x^2}\)
\(f(x)=\frac{sinx}{x}\)
\(f(x)=tan^{−1}(x)\)
a. Using the algebraic limit laws, we have
\[\lim_{x→∞}(5−\frac{2}{x^2})=\lim_{x→∞}5−2(\lim_{x→∞}\frac{1}{x}).(\lim_{x→∞}\frac{1}{x})=5−2⋅0=5.\nonumber\]
Similarly, \(\lim_{x→∞}f(x)=5\). Therefore, \(f(x)=\frac{5−2}{x^2}\) has a horizontal asymptote of \(y=5\) and \(f\) approaches this horizontal asymptote as \(x→±∞\) as shown in the following graph.
Figure \(\PageIndex{4}\): This function approaches a horizontal asymptote as \(x→±∞.\)
b. Since \(1≤sinx≤1\) for all \(x\), we have
\(\frac{−1}{x}≤\frac{sinx}{x}≤\frac{1}{x}\)
for all \(x≠0\). Also, since
\(\lim_{x→∞}\frac{−1}{x}=0=\lim_{x→∞}\frac{1}{x}\),
we can apply the squeeze theorem to conclude that
\(\lim_{x→∞}\frac{sinx}{x}=0.\)
\(\lim_{x→−∞}\frac{sinx}{x}=0.\)
Thus, \(f(x)=\frac{sinx}{x}\) has a horizontal asymptote of \(y=0\) and \(f(x)\) approaches this horizontal asymptote as \(x→±∞\) as shown in the following graph.
Figure \(\PageIndex{5}\): This function crosses its horizontal asymptote multiple times.
c. To evaluate \(\lim_{x→∞}tan^{−1}(x)\) and \(\lim_{x→−∞}tan^{−1}(x)\), we first consider the graph of \(y=tan(x)\) over the interval \((−π/2,π/2)\) as shown in the following graph.
Figure \(\PageIndex{6}\): The graph of \(tanx\) has vertical asymptotes at \(x=±\frac{π}{2}\)
\(\lim_{x→(π/2)−}tanx=∞,\)
it follows that
\(\lim_{x→∞}tan^{−1}(x)=\frac{π}{2}.\)
Similarly, since
\(\lim_{x→(π/2)^+}tanx=−∞,\)
\(\lim_{x→−∞}tan^{−1}(x)=−\frac{π}{2}.\)
As a result, \(y=\frac{π}{2}\) and \(y=−\frac{π}{2}\) are horizontal asymptotes of \(f(x)=tan^{−1}(x)\) as shown in the following graph.
Figure \(\PageIndex{7}\): This function has two horizontal asymptotes.
Exercise \(\PageIndex{1}\)
Evaluate \(\lim_{x→−∞}(3+\frac{4}{x})\) and \(\lim_{x→∞}(3+\frac{4}{x})\). Determine the horizontal asymptotes of \(f(x)=3+\frac{4}{x},\) if any.
\(\lim_{x→±∞}1/x=0\)
Both limits are \(3.\) The line \(y=3\) is a horizontal asymptote.
Sometimes the values of a function \(f\) become arbitrarily large as \(x→∞ \)(or as \(x→−∞\)). In this case, we write \(\lim_{x→∞}f(x)=∞\) (or \(\lim_{x→−∞}f(x)=∞\)). On the other hand, if the values of \(f\) are negative but become arbitrarily large in magnitude as \(x→∞\) (or as \(x→−∞\)), we write \(\lim_{x→∞}f(x)=−∞\) (or \(\lim_{x→−∞}f(x)=−∞\)).
For example, consider the function \(f(x)=x^3\). As seen in Table and Figure, as \(x→∞\) the values \(f(x)\) become arbitrarily large. Therefore, \(\lim_{x→∞}x^3=∞\). On the other hand, as \(x→−∞\), the values of \(f(x)=x3\) are negative but become arbitrarily large in magnitude. Consequently, \(\lim_{x→−∞}x^3=−∞.\)
10 20 50 100 1000
\(x^3\)
1000 8000 125,000 1,000,000 1,000,000,000
−10 −20 −50 −100 −1000
−1000 −8000 −125,000 −1,000,000 −1,000,000,000
Values of a power function as \(x→±∞\)
Figure \(\PageIndex{8}\): For this function, the functional values approach infinity as \(x→±∞.\)
Definition: infinite limit at infinity (Informal)
We say a function \(f\) has an infinite limit at infinity and write
\[\lim_{x→∞}f(x)=∞.\]
if \(f(x)\) becomes arbitrarily large for \(x\) sufficiently large. We say a function has a negative infinite limit at infinity and write
\[\lim_{x→∞}f(x)=−∞.\]
if \(f(x)<0\) and \(|f(x)|\) becomes arbitrarily large for \(x\) sufficiently large. Similarly, we can define infinite limits as \(x→−∞.\)
Earlier, we used the terms arbitrarily close, arbitrarily large, and sufficiently large to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.
Definition: limit at infinity (Formal)
We say a function \(f\) has a limit at infinity, if there exists a real number \(L\) such that for all \(ε>0\), there exists \(N>0\) such that
\[|f(x)−L|<ε\]
for all \(x>N.\) in that case, we write
\[\lim_{x→∞}f(x)=L\]
Figure \(\PageIndex{9}\): For a function with a limit at infinity, for all \(x>N, |f(x)−L|<ε.\)
Earlier in this section, we used graphical evidence in Figure and numerical evidence in Table to conclude that \(\lim_{x→∞}(\frac{2+1}{x})=2\). Here we use the formal definition of limit at infinity to prove this result rigorously.
Example \(\PageIndex{2}\):
Use the formal definition of limit at infinity to prove that \(\lim_{x→∞}(\frac{2+1}{x})=2\).
Let \(ε>0.\) Let \(N=\frac{1}{ε}\). Therefore, for all \(x>N\), we have
\[|2+\frac{1}{x}−2|=|\frac{1}{x}|=\frac{1}{x}<\frac{1}{N}=ε\]
Use the formal definition of limit at infinity to prove that \(\lim_{x→∞}(\frac{3−1}{x^2})=3\).
Let \(N=\frac{1}{\sqrt{ε}}\).
Let \(ε>0.\) Let \(N=\frac{1}{\sqrt{ε}}\). Therefore, for all \(x>N,\) we have
\(∣3−\frac{1}{x^2}−3∣=\frac{1}{x^2}<\frac{1}{N^2}=ε\)
Therefore, \(\lim_{x→∞}(3−1/x^2)=3.\)
We now turn our attention to a more precise definition for an infinite limit at infinity.
Definition: infinite limit at infinity (Formal)
\(\lim_{x→∞}f(x)=∞\)
if for all \(M>0,\) there exists an \(N>0\) such that
\(f(x)>M\)
for all \(x>N\) (see Figure).
We say a function has a negative infinite limit at infinity and write
\(\lim_{x→∞}f(x)=−∞\)
if for all \(M<0\), there exists an \(N>0\) such that
\(f(x)<M\)
for all \(x>N\).
Similarly we can define limits as \(x→−∞.\)
Figure \(\PageIndex{10}\): For a function with an infinite limit at infinity, for all \(x>N, f(x)>M.\)
Earlier, we used graphical evidence (Figure) and numerical evidence (Table) to conclude that \(\lim_{x→∞}x^3=∞\). Here we use the formal definition of infinite limit at infinity to prove that result.
Use the formal definition of infinite limit at infinity to prove that \(\lim_{x→∞}x^3=∞.\)
Let \(M>0.\) Let \(N=\sqrt[3]{M}\). Then, for all \(x>N\), we have
\(x^3>N^3=(\sqrt[3]{M})^3=M.\)
Therefore, \(\lim_{x→∞}x^3=∞\).
Use the formal definition of infinite limit at infinity to prove that \(\lim_{x→∞}3x^2=∞.\)
Let \(N=\sqrt{\frac{M}{3}}\).
Let \(M>0.\) Let \(N=\sqrt{\frac{M}{3}})\). Then, for all \(x>N,\) we have
\(3x^2>3N^2=3(\sqrt{\frac{M}{3}})^22=\frac{3M}{3}=M\)
The behavior of a function as \(x→±∞\) is called the function's \(end behavior\). At each of the function's ends, the function could exhibit one of the following types of behavior:
The function \(f(x)\) approaches a horizontal asymptote \(y=L\).
The function \(f(x)→∞\) or \(f(x)→−∞.\)
The function does not approach a finite limit, nor does it approach \(∞\) or \(−∞\). In this case, the function may have some oscillatory behavior.
Let's consider several classes of functions here and look at the different types of end behaviors for these functions.
Consider the power function \(f(x)=x^n\) where \(n\) is a positive integer. From Figure and Figure, we see that
\[\lim_{x→∞}x^n=∞;n=1,2,3,…\]
\[\lim_{x→−∞}x^n=\begin{cases}∞;7n=2,4,6,…\\−∞;&n=1,3,5,….\end{cases}\]
Figure \(\PageIndex{11}\): For power functions with an even power of \(n, \lim_{x→∞}x^n=∞=\lim_{x→−∞}x^n\).
Figure \(\PageIndex{12}\): For power functions with an odd power of \(n, \lim_{x→∞}x^n=∞\) and
\(\lim_{x→−∞}x^n=−∞.\)
Using these facts, it is not difficult to evaluate \(\lim_{x→∞}cx^n\) and \(\lim_{x→−∞}cx^n\), where \(c\) is any constant and \(n\) is a positive integer. If \(c>0\), the graph of \(y=cx^n\)is a vertical stretch or compression of \(y=x^n,\) and therefore
\(\lim_{x→∞}cx^n=\lim_{x→∞}x^n\) and \(\lim_{x→−∞}cx^n=\lim_{x→−∞}x^n\) if \(c>0\).
If \(c<0,\) the graph of \(y=cx^n\) is a vertical stretch or compression combined with a reflection about the \(x\)-axis, and therefore
\(\lim_{x→∞}cx^n=−\lim_{x→∞}x^n\) and \(\lim_{x→−∞}cx^n=−\lim_{x→−∞}x^n\) if \(c<0.\)
If \(c=0,y=cx^n=0,\) in which case \(\lim_{x→∞}cx^n=0=\lim_{x→−∞}cx^n.\)
Example \(\PageIndex{4}\): Limits at Infinity for Power Functions
For each function \(f\), evaluate \(\lim_{x→∞}f(x)\) and \(\lim_{x→−∞}f(x)\).
\(f(x)=−5x^3\)
\(f(x)=2x^4\)
Since the coefficient of \(x^3\) is \(−5\), the graph of \(f(x)=−5x^3\) involves a vertical stretch and reflection of the graph of \(y=x^3\) about the \(x\)-axis. Therefore, \(\lim_{x→∞}(−5x^3)=−∞\) and \(\lim_{x→−∞}(−5x^3)=∞\).
Since the coefficient of \(x^4\) is \(2\), the graph of \(f(x)=2x^4\) is a vertical stretch of the graph of \(y=x^4\). Therefore, \(\lim_{x→∞}2x^4=∞\) and \(\lim_{x→−∞}2x^4=∞\).
Let \(f(x)=−3x^4\). Find \(\lim_{x→∞}f(x)\).
The coefficient \(−3\) is negative.
\(−∞\)
We now look at how the limits at infinity for power functions can be used to determine \(\lim_{x→±∞}f(x)\) for any polynomial function \(f\). Consider a polynomial function
\[f(x)=a_nx^n+a_{n−1}x^n−1+…+a^1x+a^0\]
of degree \(n≥1\) so that \(a_n≠0.\)Factoring, we see that
\[f(x)=a_nx^n(1+\frac{a_{n−1}}{a_n}\frac{1}{x}+…+\frac{a_1}{a_n}\frac{1}{x^{n−1}}+\frac{a_0}{a^n}).\]
As \(x→±∞,\) all the terms inside the parentheses approach zero except the first term. We conclude that
\[\lim_{x→±∞}f(x)=\lim_{x→±∞}a_nx^n.\]
For example, the function \(f(x)=5x^3−3x^2+4\) behaves like \(g(x)=5x^3\) as \(x→±∞\) as shown in Figure and Table.
Figure \(\PageIndex{13}\): The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.
\(f(x)=5x^3−3x^2+4\)
4704 4,970,004 4,997,000,004
\(g(x)=5x^3\)
−10 −100 −000
−5296 −5,029,996 −5,002,999,996
A polynomial's end behavior is determined by the term with the largest exponent
The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In Example, we show that the limits at infinity of a rational function \(f(x)=\frac{p(x)}{q(x)}\) depend on the relationship between the degree of the numerator and the degree of the denominator. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of \(x\) appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of \(x\).
Example \(\PageIndex{5}\): Determining End Behavior for Rational Functions
For each of the following functions, determine the limits as \(x→∞\) and \(x→−∞.\) Then, use this information to describe the end behavior of the function.
\(f(x)=\frac{3x−1}{2x+5}\) (Note: The degree of the numerator and the denominator are the same.)
\(f(x)=\frac{3x^2+2x}{4x^3−5x+7}\) (Note: The degree of numerator is less than the degree of the denominator.)
\(f(x)=\frac{3x^2+4x}{x+2}\) in the denominator is x. Therefore, dividing the numerator and denominator by x and applying the algebraic limit laws, we see that
a. The highest power of \(x\) in the denominator is \(x\). Therefore, dividing the numerator and denominator by \(x\) and applying the algebraic limit laws, we see that
\(\lim_{x→±∞}\frac{3x−1}{2x+5}=\lim_{x→±∞}\frac{3−1/x}{2+5/x}\)
\(=\frac{\lim_{x→±∞}(3−1/x)}{\lim_{x→±∞}(2+5/x)}\)
\(=\frac{\lim_{x→±∞}3−\lim_{x→±∞}1/x}{\lim_{x→±∞}2+\lim_{x→±∞}5/x}\)
\(=\frac{3−0}{2+0}=\frac{3}{2}.\)
Since \(\lim_{x→±∞}f(x)=\frac{3}{2}\), we know that \(y=\frac{3}{2}\) is a horizontal asymptote for this function as shown in the following graph.
Figure \(\PageIndex{14}\): The graph of this rational function approaches a horizontal asymptote as \(x→±∞.\)
b. Since the largest power of \(x\) appearing in the denominator is \(x^3\), divide the numerator and denominator by \(x^3\). After doing so and applying algebraic limit laws, we obtain
\(\lim_{x→±∞}\frac{3x^2+2x}{4x^3−5x+7}=\lim_{x→±∞}\frac{3/x+2/x^2}{4−5/x^2+7/x^3}=\frac{3.0+2.0}{4−5.0+7.0}=0.\)
Therefore \(f\) has a horizontal asymptote of \(y=0\) as shown in the following graph.
Figure \(\PageIndex{15}\): The graph of this rational function approaches the horizontal asymptote \(y=0\) as \(x→±∞.\)
c. Dividing the numerator and denominator by \(x\), we have
\(\lim_{x→±∞}\frac{3x^2+4x}{x+2}=\lim_{x→±∞}\frac{3x+4}{1+2/x}\).
As \(x→±∞\), the denominator approaches \(1\). As \(x→∞\), the numerator approaches \(+∞\). As \(x→−∞\), the numerator approaches \(−∞\). Therefore \(\lim_{x→∞}f(x)=∞\), whereas \(\lim_{x→−∞}f(x)=−∞\) as shown in the following figure.
Figure \(\PageIndex{16}\): As \(→∞\), the values \(f(x)→∞\). As \(x→−∞\), the values \(f(x)→−∞.\)
Evaluate \(\lim_{x→±∞}\frac{3x^2+2x−1}{5x^2−4x+7}\) and use these limits to determine the end behavior of \(f(x)=\frac{3x^2+2x−2}{5x^2−4x+7}\).
Divide the numerator and denominator by \(x^2\).
\(\frac{3}{5}\)
Before proceeding, consider the graph of \(f(x)=\frac{(3x^2+4x)}{(x+2)}\) shown in Figure. As \(x→∞\) and \(x→−∞\), the graph of \(f\) appears almost linear. Although \(f\) is certainly not a linear function, we now investigate why the graph of \(f\) seems to be approaching a linear function. First, using long division of polynomials, we can write
\(f(x)=\frac{3x^2+4x}{x+2}=3x−2+\frac{4}{x+2}\).
Since \(\frac{4}{(x+2)}→0\) as \(x→±∞,\) we conclude that
\(\lim_{x→±∞}(f(x)−(3x−2))=\lim_{x→±∞}\frac{4}{x+2}=0.\)
Therefore, the graph of \(f\) approaches the line \)y=3x−2\) as \(x→±∞\). This line is known as an oblique asymptote for \(f\) (Figure).
Figure \(\PageIndex{17}\): The graph of the rational function \(f(x)=(3x^2+4x)/(x+2)\) approaches the oblique asymptote \(y=3x−2\) as \(x→±∞.\)
We can summarize the results of Example to make the following conclusion regarding end behavior for rational functions. Consider a rational function
\(f(x)=\frac{p(x)}{q(x)}=\frac{a_nx^n+a_{n−1}x^{n−1}+…+a_1x+a_0}{b_mx^m+b_{m−1}x^{m−1}+…+b_1x+b_0}\),
where \(a_n≠0\) and \(b_m≠0.\)
1. If the degree of the numerator is the same as the degree of the denominator \((n=m),\) then \(f\) has a horizontal asymptote of \(y=a_n/b_m\) as \(x→±∞.\)
2. If the degree of the numerator is less than the degree of the denominator \((n<m),\) then \(f\) has a horizontal asymptote of \(y=0\) as \(x→±∞.\)
3. If the degree of the numerator is greater than the degree of the denominator \((n>m),\) then \(f\) does not have a horizontal asymptote. The limits at infinity are either positive or negative infinity, depending on the signs of the leading terms. In addition, using long division, the function can be rewritten as
\(f(x)=\frac{p(x)}{q(x)}=g(x)+\frac{r(x)}{q(x)}\),
where the degree of \(r(x)\) is less than the degree of \(q(x)\). As a result, \(\lim_{x→±∞}r(x)/q(x)=0\). Therefore, the values of \([f(x)−g(x)]\) approach zero as \(x→±∞\). If the degree of \(p(x)\) is exactly one more than the degree of \(q(x) (n=m+1)\), the function \(g(x)\) is a linear function. In this case, we call \(g(x)\) an oblique asymptote.
Now let's consider the end behavior for functions involving a radical.
Example \(\PageIndex{6}\): Determining End Behavior for a Function Involving a Radical
Find the limits as \(x→∞\) and \(x→−∞\) for \(f(x)=\frac{3x−2}{\sqrt{4x^2+5}}\) and describe the end behavior of \(f\).
Let's use the same strategy as we did for rational functions: divide the numerator and denominator by a power of \(x\). To determine the appropriate power of \(x\), consider the expression \(\sqrt{4x^2+5}\) in the denominator. Since
\[\sqrt{4x^2+5}≈\sqrt{4x^2}=2|x|\]
for large values of \(x\) in effect \(x\) appears just to the first power in the denominator. Therefore, we divide the numerator and denominator by \(|x|\). Then, using the fact that \(|x|=x\) for \(x>0, |x|=−x\) for \(x<0\), and \(|x|=\sqrt{x^2}\) for all \(x\), we calculate the limits as follows:
\(\lim_{x→∞}\frac{3x−2}{\sqrt{4x^2+5}}=\lim_{x→∞}\frac{(1/|x|)(3x−2)}{(1/|x|)\sqrt{4x^2+5}}\)
\(=\lim_{x→∞}\frac{(1/x)(3x−2)}{\sqrt{(1/x^2)(4x2+5)}}\)
\(=\lim_{x→∞}\frac{3−2/x}{\sqrt{4+5/x^2}}=\frac{3}{\sqrt{4}}=\frac{3}{2}\)
\(\lim_{x→−∞}\frac{3x−2}{\sqrt{4x^2+5}}=\lim_{x→−∞}\frac{(1/|x|)(3x−2)}{(1/|x|)\sqrt{4x^2+5}}\)
\(=\lim_{x→−∞}\frac{(−1/x)(3x−2)}{\sqrt{(1/x^2)(4x^2+5)}}\)
\(=\lim_{x→−∞}\frac{−3+2/x}{\sqrt{4+5/x^2}}=\frac{−3}{\sqrt{4}}=\frac{−3}{2}\).
Therefore, \(f(x)\) approaches the horizontal asymptote \(y=\frac{3}{2}\) as \(x→∞\) and the horizontal asymptote \(y=−\frac{3}{2}\) as \(x→−∞\) as shown in the following graph.
Figure \(\PageIndex{18}\):This function has two horizontal asymptotes and it crosses one of the asymptotes.
Evaluate \(\lim_{x→∞}\frac{\sqrt{3x^2+4}}{x+6}\).
Divide the numerator and denominator by \(|x|.\)
\(±\sqrt{3}\)
The six basic trigonometric functions are periodic and do not approach a finite limit as \(x→±∞.\) For example, \(sinx\) oscillates between \(1and−1\) (Figure). The tangent function \(x\) has an infinite number of vertical asymptotes as \(x→±∞\); therefore, it does not approach a finite limit nor does it approach \(±∞\) as \(x→±∞\) as shown in Figure.
Figure \(\PageIndex{19}\):The function \(f(x)=sinx\) oscillates between \(1and−1\) as \(x→±∞\)
Figure \(\PageIndex{20}\): The function \(f(x)=tanx\) does not approach a limit and does not approach \(±∞\) as \(x→±∞\)
Recall that for any base \(b>0,b≠1,\) the function \(y=b^x\) is an exponential function with domain \((−∞,∞)\) and range \((0,∞)\). If \(b>1,y=b^x\) is increasing over `\((−∞,∞)\). If \(0<b<1, y=b^x\) is decreasing over \((−∞,∞).\) For the natural exponential function \(f(x)=e^x, e≈2.718>1\). Therefore, \(f(x)=e^x\) is increasing on `\((−∞,∞)\) and the range is `\((0,∞).\) The exponential function \(f(x)=e^x\) approaches \(∞\) as \(x→∞\) and approaches \(0\) as \(x→−∞\) as shown in Table and Figure.
End behavior of the natural exponential function
−5 −2 0 2 5
\(e^x\)
0.00674 0.135 1 7.389 148.413
Figure \(\PageIndex{21}\): The exponential function approaches zero as \(x→−∞\) and approaches \(∞\) as \(x→∞.\)
Recall that the natural logarithm function \(f(x)=ln(x)\) is the inverse of the natural exponential function \(y=e^x\). Therefore, the domain of \(f(x)=ln(x)\) is \((0,∞)\) and the range is \((−∞,∞)\). The graph of \(f(x)=ln(x)\) is the reflection of the graph of \(y=e^x\) about the line \(y=x\). Therefore, \(ln(x)→−∞\) as \(x→0^+\) and \(ln(x)→∞\) as \(x→∞\) as shown in Figure and Table.
End behavior of the natural logarithm function
0.01 0.1 1 10 100
\(ln(x)\)
−4.605 −2.303 0 2.303 4.605
Figure \(\PageIndex{22}\): The natural logarithm function approaches \(∞\) as \(x→∞.\)
Example \(\PageIndex{7}\): Determining End Behavior for a Transcendental Function
Find the limits as \(x→∞\) and \(x→−∞\) for \(f(x)=\frac{(2+3e^x)}{(7−5ex^)}\) and describe the end behavior of \(f.\)
To find the limit as \(x→∞,\) divide the numerator and denominator by \(e^x\):
\(\lim_{x→∞}f(x)=\lim_{x→∞}\frac{2+3e^x}{7−5e^x}\)
\(=\lim_{x→∞}\frac{(2/e^x)+3}{(7/e^x)−5.}\)
As shown in Figure, \(e^x→∞\) as \(x→∞.\) Therefore,
\(\lim_{x→∞}\frac{2}{e^x}=0=\lim_{x→∞}\frac{7}{e^x}\).
We conclude that \(\lim_{x→∞f}(x)=−\frac{3}{5}\), and the graph of \(f\) approaches the horizontal asymptote \(y=−\frac{3}{5}\) as \(x→∞.\) To find the limit as \(x→−∞\), use the fact that \(e^x→0\) as \(x→−∞\) to conclude that \(\lim_{x→∞}f(x)=\frac{2}{7}\), and therefore the graph of approaches the horizontal asymptote \(y=\frac{2}{7}\) as \(x→−∞\).
Find the limits as \(x→∞\) and \(x→−∞\) for \(f(x)=\frac{(3e^x−4)}{(5e^x+2).}\)
\(\lim_{x→∞}e^x=∞\) and \(im_{x→∞}e^x=0.\)
\(\lim_{x→∞}f(x)=\frac{3}{5}, \lim_{x→−∞}f(x)=−2\)
We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let's look at a general strategy to use when graphing any function.
Problem-Solving Strategy: Drawing the Graph of a Function
Given a function \(f\), use the following steps to sketch a graph of \(f\):
Determine the domain of the function.
Locate the \(x\)- and \(y\)-intercepts.
Evaluate \(\lim_{x→∞}f(x)\) and \(\lim_{x→−∞}f(x)\) to determine the end behavior. If either of these limits is a finite number \(L\), then \(y=L\) is a horizontal asymptote. If either of these limits is \(∞\) or \(−∞\), determine whether \(f\) has an oblique asymptote. If \(i\)s a rational function such that \(f(x)=\frac{p(x)}{q(x)}\), where the degree of the numerator is greater than the degree of the denominator, then \(f\) can be written as \[f(x)=\frac{p(x)}{q(x)}=g(x)+\frac{r(x)}{q(x),}\] where the degree of \(r(x)\) is less than the degree of \(q(x)\). The values of \(f(x)\) approach the values of \(g(x)\) as \(x→±∞\). If \(g(x)\) is a linear function, it is known as an oblique asymptote.
Determine whether \(f\) has any vertical asymptotes.
Calculate \(f′.\) Find all critical points and determine the intervals where \(f\) is increasing and where \(f\) is decreasing. Determine whether \(f\) has any local extrema.
Calculate \(f''.\) Determine the intervals where \(f\) is concave up and where \(f\) is concave down. Use this information to determine whether \(f\) has any inflection points. The second derivative can also be used as an alternate means to determine or verify that \(f\) has a local extremum at a critical point.
Now let's use this strategy to graph several different functions. We start by graphing a polynomial function.
Example \(\PageIndex{8}\): Sketching a Graph of a Polynomial
Sketch a graph of \(f(x)=(x−1)^2(x+2).\)
Step 1: Since \(f\) is a polynomial, the domain is the set of all real numbers.
Step 2: When \(x=0,f(x)=2.\) Therefore, the y-intercept is \((0,2)\). To find the \(x\)-intercepts, we need to solve the equation \((x−1)^2(x+2)=0\), gives us the \(x\)-intercepts \((1,0)\) and \((−2,0)\)
Step 3: We need to evaluate the end behavior of \(f.\)As \(x→∞, (x−1)^2→∞\) and \((x+2)→∞\). Therefore, \(\lim_{x→∞}f(x)=∞\) As \(x→−∞, (x−1)^2→∞\) and \((x+2)→−∞\). Therefore, \(\lim_{x→∞}f(x)=−∞\). To get even more information about the end behavior of \(f\), we can multiply the factors of \(f\).When doing so, we see that
\[f(x)=(x−1)^2(x+2)=x^3−3x+2.\]
Since the leading term of \(f\) is \(x^3\), we conclude that \(f\) behaves like \(y=x^3\) as \(x→±∞.\)
Step 4: Since \(f\) is a polynomial function, it does not have any vertical asymptotes.
Step 5: The first derivative of \(f\) is
\[f′(x)=3x^2−3.\]
Therefore, \(f\) has two critical points: \(x=1,−1.\) Divide the interval \((−∞,∞)\) into the three smaller intervals: \((−∞,−1), (−1,1)\), and \((1,∞)\). Then, choose test points \(x=−2, x=0\), and \(x=2\) from these intervals and evaluate the sign of \(f′(x)\) at each of these test points, as shown in the following table.
Sign of Derivative \(f'(x)=3x^2−3=3(x−1)(x+1)\)
\((−∞,−1)\) \(x=−2\) \((+)(−)(−)=+\) \(f\) is increasing
\((−1,1)\) \(x=0\) \((+)(−)(+)=−\) \(f\) decreasing
\((1,∞)\) (x=2\) \((+)(+)(+)=+\) \(f\) is increasing
From the table, we see that \(f\) has a local maximum at \(x=−1\) and a local minimum at \(x=1\). Evaluating \(f(x)\) at those two points, we find that the local maximum value is \(f(−1)=4\) and the local minimum value is \(f(1)=0.\)
Step 6: The second derivative of \(f\) is
\[f''(x)=6x.\]
The second derivative is zero at \(x=0.\) Therefore, to determine the concavity of \(f\), divide the interval \((−∞,∞)\) into the smaller intervals \((−∞,0)\) and \((0,∞)\), and choose test points \(x=−1\) and \(x=1\) to determine the concavity of \(f\) on each of these smaller intervals as shown in the following table.
Sign of \(''(x)=6x\)
\((−∞,0)\) \(x=−1\) \(−\) \(f\) is concave down..
\((0,∞)\) \(x=1\) \(+\) \(f\) is concave up.
We note that the information in the preceding table confirms the fact, found in step \(5\), that f has a local maximum at \(x=−1\) and a local minimum at \(x=1\). In addition, the information found in step \(5\)—namely, \(f\) has a local maximum at \(x=−1\) and a local minimum at \(x=1\), and \(f′(x)=0\) at those points—combined with the fact that \(f''\) changes sign only at \(x=0\) confirms the results found in step \(6\) on the concavity of \(f\).
Combining this information, we arrive at the graph of \(f(x)=(x−1)^2(x+2)\) shown in the following graph.
\(f\) is a fourth-degree polynomial.
Example \(\PageIndex{9}\): Sketching a Rational Function
Sketch the graph of \(f(x)=\frac{x^2}{(1−x^2)}\).
Step 1: The function \(f\) is defined as long as the denominator is not zero. Therefore, the domain is the set of all real numbers \(x\) except \(x=±1.\)
Step 2: Find the intercepts. If \(x=0,\) then \(f(x)=0\), so \(0\) is an intercept. If \(y=0\), then \(\frac{x^2}{(1−x^2)}=0,\) which implies x=0. Therefore, (0,0) is the only intercept.
Step 3: Evaluate the limits at infinity. Since \(f\)is a rational function, divide the numerator and denominator by the highest power in the denominator: \(x^2\).We obtain
\(\lim_{x→±∞}\frac{x^2}{1−x^2}=\lim_{x→±∞}\frac{1}{\frac{1}{x^2}−1}=−1.\)
Therefore, \(f\) has a horizontal asymptote of \(y=−1\) as \(x→∞\) and \(x→−∞.\)
Step 4: To determine whether \(f\) has any vertical asymptotes, first check to see whether the denominator has any zeroes. We find the denominator is zero when \(x=±1\). To determine whether the lines \(x=1\) or \(x=−1\) are vertical asymptotes of \(f\), evaluate \(\lim_{x→1}f(x)\) and \(\lim_{x→−1}f(x)\). By looking at each one-sided limit as \(x→1,\) we see that
\(\lim_{x→1^+}\frac{x^2}{1−x^2}=−∞\) and \(\lim_{x→1^−}\frac{x^2}{1−x^2}=∞.\)
In addition, by looking at each one-sided limit as \(x→−1,\) we find that
\(\lim_{x→−1^+}\frac{x^2}{1−x^2}=∞\) and \(\lim_{x→−1^−}\frac{x^2}{1−x^2}=−∞.\)
Step 5: Calculate the first derivative:
\(f′(x)=\frac{(1−x^2)(2x)−x^2(−2x)}{(1−x^2)^2}=\frac{2x}{(1−x^2)^2}\).
Critical points occur at points \(x\) where \(f′(x)=0\) or \(f′(x)\) is undefined. We see that \(f′(x)=0\) when \(x=0.\) The derivative \(f′\) is not undefined at any point in the domain of \(f\). However, \(x=±1\) are not in the domain of \(f\). Therefore, to determine where \(f\) is increasing and where \(f\) is decreasing, divide the interval \((−∞,∞)\) into four smaller intervals: \((−∞,−1), (−1,0), (0,1),\) and \((1,∞)\), and choose a test point in each interval to determine the sign of \(f′(x)\) in each of these intervals. The values \(x=−2, x=−\frac{1}{2}, x=\frac{1}{2}\), and \(x=2\) are good choices for test points as shown in the following table.
Sign of \(f′(x)=\frac{2x}{(1−x^2)^2}\)
\((−∞,−1)\) \(x=−2\) \(−/+=−\) \(f\) is decreasing.
\((−1,0)\) \(x=−/2\) \(−/+=−\) \(f\) is decreasing.
\((0,1)\) \(x=1/2\) \(+/+=+\) \(f\) is increasing.
\((1,∞)\) \(x=2\) \(+/+=+\) \(f\) is increasing.
From this analysis, we conclude that \(f\) has a local minimum at \(x=0\) but no local maximum.
Step 6: Calculate the second derivative:
\(f''(x)=\frac{(1−x^2)^2(2)−2x(2(1−x^2)(−2x))}{(1−x^2)^4}\)
\(=\frac{(1−x^2)[2(1−x^2)+8x^2]}{(1−x^2)^4}\)
\(=\frac{2(1−x^2)+8x^2}{(1−x^2)^3}\)
\(=\frac{6x^2+2}{(1−x^2)^3}.\)
To determine the intervals where \(f\) is concave up and where \(f\) is concave down, we first need to find all points \(x\) where \(f''(x)=0\) or \(f''(x)\) is undefined. Since the numerator \(6x^2+2≠0\) for any \(x, f''(x)\) is never zero. Furthermore, \(f''\) is not undefined for any \(x\) in the domain of \(f\). However, as discussed earlier, \(x=±1\) are not in the domain of \(f\). Therefore, to determine the concavity of \(f\), we divide the interval \((−∞,∞)\) into the three smaller intervals \((−∞,−1), (−1,−1)\), and \((1,∞)\), and choose a test point in each of these intervals to evaluate the sign of \(f''(x)\). in each of these intervals. The values \(x=−2, x=0\), and \(x=2\) are possible test points as shown in the following table.
Sign of \(f''(x)=\frac{6x^2+2}{(1−x^2)^3}\)
\((−∞,−1)\) \(x=−2\) \(+/−=−\) \(f\) is concave down.
\((−1,−1)\) \(x=0\) \(+/+=+\) \(f\) is concave up
\((1,∞)\) \(x=2\) \(+/−=−\) \(f\) is concave down.
Combining all this information, we arrive at the graph of \(f\) shown below. Note that, although \(f\) changes concavity at \(x=−1\) and \(x=1\), there are no inflection points at either of these places because \(f\) is not continuous at \(x=−1\) or \(x=1.\)
<iframe src="https://www.desmos.com/calculator/rkgoh8veqg?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>
Sketch a graph of \(f(x)=\frac{(3x+5)}{(8+4x).}\)
A line \(y=L\) is a horizontal asymptote of \(f\) if the limit as \(x→∞\) or the limit as \(x→−∞\) of \(f(x)\) is \(L\). A line \(x=a\) is a vertical asymptote if at least one of the one-sided limits of \(f\) as \(x→a\) is \(∞\) or \(−∞.\)
Example \(\PageIndex{10}\): Sketching a Rational Function with an Oblique Asymptote
Sketch the graph of \(f(x)=\frac{x^2}{(x−1)}\)
Step 1: The domain of \(f\) is the set of all real numbers \(x\) except \(x=1.\)
Step 2: Find the intercepts. We can see that when \(x=0, f(x)=0,\) so \((0,0)\) is the only intercept.
Step 3: Evaluate the limits at infinity. Since the degree of the numerator is one more than the degree of the denominator, \(f\) must have an oblique asymptote. To find the oblique asymptote, use long division of polynomials to write
\(f(x)=\frac{x^2}{x−1}=x+1+\frac{1}{x−1}\).
Since \(1/(x−1)→0\) as \(x→±∞, f(x)\) approaches the line \(y=x+1\) as \(x→±∞\). The line \(y=x+1\) is an oblique asymptote for \(f\).
Step 4: To check for vertical asymptotes, look at where the denominator is zero. Here the denominator is zero at \(x=1.\) Looking at both one-sided limits as \(x→1,\) we find
\(\lim_{x→1^+}\frac{x^2}{x−1}=∞\) and \(\lim_{x→1^−}\frac{x^2}{x−1}=−∞.\)
Therefore, \(x=1\) is a vertical asymptote, and we have determined the behavior of \(f\) as \(x\) approaches \(1\) from the right and the left.
\(f′(x)=\frac{(x−1)(2x)−x^2(1)}{(x−1)^2}=\frac{x^2−2x}{(x−1)^2}.\)
We have \(f′(x)=0\) when \(x^2−2x=x(x−2)=0\). Therefore, \(x=0\) and \(x=2\) are critical points. Since \(f\) is undefined at \(x=1\), we need to divide the interval \((−∞,∞)\) into the smaller intervals \((−∞,0), (0,1), (1,2),\) and \((2,∞)\), and choose a test point from each interval to evaluate the sign of \(f′(x)\) in each of these smaller intervals. For example, let \(x=−1, x=\frac{1}{2}, x=\frac{3}{2}\), and \(x=3\) be the test points as shown in the following table.
Sign of \(f''(x)=\frac{2}{(x−1)^3}\)
\((−∞,1)\) \(x=0\) \(+/−=−\) \(f\) is concave down.
\((1,∞)\) (x=2\) \(+/+=+\) \(f\) is concave up
From the information gathered, we arrive at the following graph for \(f.\)
Exercise \(\PageIndex{10}\)
Find the oblique asymptote for \(f(x)=\frac{(3x^3−2x+1)}{(2x^2−4)}\).
Use long division of polynomials.
\(y=\frac{3}{2}x\)
Example \(\PageIndex{11}\): Sketching the Graph of a Function with a Cusp
Sketch a graph of \((x)=(x−1)^{2/3}\)
Step 1: Since the cube-root function is defined for all real numbers \(x\) and \((x−1)^{2/3}=(\sqrt[3]{x−1})^2\), the domain of \(f\) is all real numbers.
Step 2: To find the \(y\)-intercept, evaluate \(f(0)\). Since \(f(0)=1,\) the \(y\)-intercept is \((0,1)\). To find the \(x\)-intercept, solve \((x−1)^{2/3}=0\). The solution of this equation is \(x=1\), so the \(x\)-intercept is \((1,0).\)
Step 3: Since \(\lim_{x→±∞}(x−1)^{2/3}=∞,\) the function continues to grow without bound as \(x→∞\) and \(x→−∞.\)
Step 4: The function has no vertical asymptotes.
Step 5: To determine where \(f\) is increasing or decreasing, calculate \(f′.\) We find
\[f′(x)=\frac{2}{3}(x−1)^{−1/3}=\frac{2}{3(x−1)^{1/3}}\]
This function is not zero anywhere, but it is undefined when \(x=1.\) Therefore, the only critical point is \(x=1.\) Divide the interval \((−∞,∞)\) into the smaller intervals \((−∞,1)\) and \((1,∞)\), and choose test points in each of these intervals to determine the sign of \(f′(x)\) in each of these smaller intervals. Let \(x=0\) and \(x=2\) be the test points as shown in the following table.
Sign of \(f′(x)=\frac{2}{3(x−1)^{1/3}}\)
\((−∞,1)\) \(x=0\) \(+/−=−\) \(f\) is decreasing
\((1,∞)\) \(x=2\) \(+/+=+\) \(f\) is increasing
We conclude that \(h\)as a local minimum at \(x=1\). Evaluating \(f\) at \(x=1\), we find that the value of \(f\) at the local minimum is zero. Note that \(f′(1)\) is undefined, so to determine the behavior of the function at this critical point, we need to examine \(\lim_{x→1}f′(x).\) Looking at the one-sided limits, we have
\[\lim_{x→1^+}\frac{2}{3(x−1)^{1/3}}=∞\) and \(\lim_{x→1^−}\frac{2}{3(x−1)^{1/3}}=−∞.\]
Therefore, \(f\) has a cusp at \(x=1.\)
Step 6: To determine concavity, we calculate the second derivative of \(f:\)
\[f''(x)=−\dfrac{2}{9}(x−1)^{−4/3}=\dfrac{−2}{9(x−1)^{4/3}}.\]
We find that \(f''(x)\) is defined for all \(x\), but is undefined when \(x=1\). Therefore, divide the interval \((−∞,∞)\) into the smaller intervals \((−∞,1)\) and \((1,∞)\), and choose test points to evaluate the sign of \(f''(x)\) in each of these intervals. As we did earlier, let \(x=0\) and \(x=2\) be test points as shown in the following table.
Sign of \(f''(x)=\frac{−2}{3(x−1)^{4/3}}\)
\((−∞,1)\) \(x=0\) \(−/+=−\) \(f\) is concave down
\((1,∞)\) \(x=2\) \−(+/+=−\) \(f\) is concave down
From this table, we conclude that \(i\)s concave down everywhere. Combining all of this information, we arrive at the following graph for \(f\).
Consider the function \(f(x)=5−x^{2/3}\). Determine the point on the graph where a cusp is located. Determine the end behavior of \(f\).
A function \(f\) has a cusp at a point a if \(f(a)\) exists, \(f'(a)\) is undefined, one of the one-sided limits as \(x→a\) of \(f'(x) is +∞\), and the other one-sided limit is \(−∞.\)
The function \(f\) has a cusp at \((0,5) \lim_{x→0^−}f′(x)=∞\), \(\lim_{x→0^+}f′(x)=−∞\). For end behavior, \(\lim_{x→±∞}f(x)=−∞.\)
The limit of \(f(x)\) is \(L\) as \(x→∞\) (or as \(x→−∞)\) if the values \(f(x)\) become arbitrarily close to \(L\) as \(x\)becomes sufficiently large.
The limit of \(f(x)\) is \(∞\) as \(x→∞\) if \(f(x)\) becomes arbitrarily large as \(x\) becomes sufficiently large. The limit of \(f(x)\) is \(−∞\) as \(x→∞\) if \(f(x)<0\) and \(|f(x)|\) becomes arbitrarily large as \(x\) becomes sufficiently large. We can define the limit of \(f(x)\) as \(x\) approaches \(−∞\) similarly.
For a polynomial function \(p(x)=a_nx^n+a_{n−1}x^{n−1}+…+a_1x+a_0,\) where \(a_n≠0\), the end behavior is determined by the leading term \(a_nx^n\). If \(n≠0, p(x)\) approaches \(∞\) or \(−∞\)at each end.
For a rational function \(f(x)=\frac{p(x)}{q(x),}\) the end behavior is determined by the relationship between the degree of \(p\) and the degree of \(q\). If the degree of \(p\) is less than the degree of \(q\), the line \(y=0\) is a horizontal asymptote for \(f\). If the degree of \(p\) is equal to the degree of \(q\), then the line \(y=\frac{a_n}{b_n}\) is a horizontal asymptote, where \(a_n\) and \(b_n\) are the leading coefficients of \(p\) and \(q\), respectively. If the degree of \(p\) is greater than the degree of \(q\), then \(f\) approaches \(∞\) or \(−∞\) at each end.
the behavior of a function as \(x→∞\) and \(x→−∞\)
horizontal asymptote
if \(\lim_{x→∞}f(x)=L\) or \(\lim_{x→−∞}f(x)=L\), then \(y=L\) is a horizontal asymptote of \(f\)
infinite limit at infinity
a function that becomes arbitrarily large as x becomes large
limit at infinity
a function that becomes arbitrarily large as \(x\) becomes large
oblique asymptote
the line \(y=mx+b\) if \(f(x)\) approaches it as \(x→∞\) or\( x→−∞\)
Gilbert Strang (MIT) and Edwin "Jed" Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.
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4.E: Applications of Derivatives (ALL Chap 4 Exercises)
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Winners for all Competitions (Chess, Carrom, Essay writing, Video games, Art, Sudoku , Debate , Vocabulary , Geography , Math bee) will be given Trophies at 6:45 PM after Dhoop Aarti on Saturday March 16 th , 2019.
Geography Bee we will be having groups of 1st -2nd grade, 3rd -4th grade, and 5th -6th grade. 20 multiple choice questions and also 5 tie breaker questions will be given. If we do not have enough kids enrolled in a particular group, we might cancel that group. Geography bee starts on Saturday March 16 th 2019 12:30 PM for 1 hr.
For topics for Grade 1st to 6th some of the study links that students can follow for preparation.
Vocabulary Bee Contest will group 1st -2nd grade, 3rd -4th grade, and 5th -6th grade. 20 multiple choice questions and 5 tie breaker questions will be given. Vocabulary Contest starts on Saturday March 16 th 2019 at 2:30PM for 1 hr. Pick correct attached document for vocabulary bee for your child grade level.
NESSP Math Bee will be based on Math Kangaroo syllabus. Every grade level will get 30 multiple choice questions. The first 25 questions will be used to select the winners. The last 5 questions will be used only if there is a tie. Students will answer all 30 questions in 30 minutes. Math Bee would start on Saturday March 16 th 2019 at 4:30PM for 1 hr.
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CAAC ATPL Written Test Study Material for 2019
We are pleased to released the most latest CAAC ATPL Written Test Question Bank
Published Date: Jul/29/2019
On the links below, you will find a wide range of questions which will help you to prepare for the CAAC written ATP exam. Once you are approved for an interview, we also have an online study tool which will provide simulated tests to check your proficiency level.
Chapter 1 - CCAR (Questions) / Chapter 1 - CCAR (Answers)
Chapter 2 - Navigational Information, Navigational Notice and Advisory Materials
Chapter 3 - Meteorology
Chapter 4 - Aerodynamics, Aircraft Flight Principles, Inspection and Operation
Chapter 5 - Navigation
Chapter 6 - Aircraft safety, Efficient Operation and Airspace System
Chapter 6E - Aircraft safety, Efficient Operation and Airspace System
Chapter 7 - Weight and Balance
Chapter 7E - Weight and Balance
Chapter 8 - Flight Performance
Chapter 9 - Air Cargo Transport Procedures
Chapter 10 - Human Factor
***Note: You will be requestd to Login or Sign Up at www.WASINC.net at no cost or obligation in order to download the above documents.***
Chinese airlines court Cathay pilots facing pay squeeze
SHANGHAI/SINGAPORE (Reuters) - Chinese airlines are circling disaffected pilots at Hong Kong's Cathay Pacific Airways (0293.HK), offering sky-high salaries to fill a shortage of experienced captains in a rapidly expanding aviation industry.
Published Date: Nov/25/2017
Brenda Goh, Jamie Freed
Dave Ross, President and CEO of WASINC International, poses beside a recruitment banner in Hong Kong, China November 20, 2017. Picture taken November 20, 2017. REUTERS/Bobby Yip
Chinese carriers, including China Southern Airlines (600029.SS), Hainan Airlines (600221.SS) and Juneyao Airlines (603885.SS), will need more than 100,000 new pilots over the next two decades, according to forecasts by planemaker Boeing Co (BA.N), as air travel takes off with increased prosperity in the world's second-largest economy.
China trains around 4,700 new pilots a year at home and abroad, according to the Civil Aviation Administration of China (CAAC), but the industry's sharp growth has left it short of captains with several thousand hours flying experience.
"Chinese airlines are growing so quickly, and ... it takes a lot of time to become a captain," said Xu Dandi, head of recruitment at Yu Feng Consulting, whose clients include China Southern and Loong Air. "Pilots hired in Hong Kong have likely passed strict assessments ... and have experience flying in Asia."
Pilot recruitment firms have flocked to Hong Kong hoping to attract established pilots from loss-making Cathay, which has this year axed 600 jobs in its biggest round of staff cuts in almost 20 years.
While Cathay is not firing any of its 2,500 Hong Kong-based pilots, it has warned of cuts next month to their housing allowances, which for some veteran captains can be worth up to HK$100,000 (£9621.38) a month.
Some Cathay pilots - who are widely regarded as among the industry's best-qualified and highest-paid - say cuts could price them out of living in one of the world's most expensive cities. Many are expatriate Australians, Americans and Britons.
"Most expat pilots will have a serious re-think ... if the (housing) allowance is cut too drastically," said one of nine Cathay captains interviewed by Reuters - none of whom wanted to be named because of the sensitivity of the matter.
Some Chinese airlines are offering tax-free pay packages of up to $314,000 a year, according to job ads. That's roughly in line with what Cathay captains can earn now, before any allowance cuts. The airline is also looking to reduce pensions and increase productivity.
Chinese airlines use recruitment agencies to scout globally for pilots. Hong Kong was not previously a major target, but has come into focus as Cathay has struggled.
Longreach Aviation Resources told Hong Kong media it received inquiries from 200 Cathay pilots before it abruptly canceled a roadshow last month, citing legal advice. It declined further comment.
Another recruiter, WASINC International, attracted nine Cathay pilots on the first day of a Hong Kong roadshow this week, and Yu Feng Consulting said it plans a similar event based on interest from mainland Chinese airlines.
"We watch for airlines that are having difficulties. Cathay has always been very strong and now they're downgrading their pay a little, a lot of their pilots are worried," WASINC CEO Dave Ross said. "(Cathay pilots) have a reputation for being high-quality pilots, so a lot of airlines are interested in them."
Cathay Pacific Chief Operations and Service Delivery Officer Greg Hughes said the airline was used to having its pilots courted by rival carriers.
"For 30-40 years, people have targeted our pilots and we don't expect that to let up," he told Reuters. "People like working for us, and our (staff) turnover rates are low."
FOREIGN CAPTAINS NEEDED
As China's airline sector has ballooned, it's had to import foreign experience, with the number of foreign pilots flying with Chinese airlines doubling to more than 1,000 between 2010 and last year.
In September, the CAAC relaxed some of the industry's strictest medical requirements - seen as a sign of how serious the pilot shortage is.
While Chinese carriers may offer top dollar packages, some foreign pilots who have moved to China warn of trade-offs.
Four expatriate pilots told Reuters the downsides include a long hiring process; short, 3-year contracts, with no union protection; gruel ling flight rosters handed to pilots at late notice; and a system geared towards penalizing pilots for mistakes rather than using these to learn safety lessons.
"The money's definitely good, but people considering the job - if that's their primary motivation - I don't think they'll last long," said Mike Edelman, an American pilot who has flown for Juneyao Airlines for nearly eight years.
"You've got to devote a lot more hours to work and ... if you want to last in China you've got to be willing to adapt to the culture."
The cultural challenge was acknowledged by one of the Cathay pilots attending the WASINC recruitment session.
"In Hong Kong, you know what's going on ... but working in China, it's a foreign country. There are issues like moving money out, financial security and pollution. Those aside, China is now better placed in terms of contracts than 10 years ago."
China isn't alone in trying to attract pilots.
Airlines in Australia, Europe and North America are again recruiting after several quiet years, and could prove an attractive alternative for some - although the seniority system used by most Western carriers, where new joiners enter on the bottom rank and salary rungs, whatever their prior experience - would likely deter Cathay's senior captains.
Of the nine Cathay captains Reuters spoke to, some named non-Chinese carriers including Vietnam Airlines HVN.HNO and Korean Air Lines Co Ltd (003490.KS) as potential options, while others said they would consider non-flying jobs.
Three captains said joining China Southern could be an option as it has bases in countries like Australia, where pilots can return home and fly to and from the carrier's Guangzhou hub, though the pay would be less than if they moved to China.
"The money isn't everything," one said.
"The attraction of living in their home country, having family away from the pollution and children growing up back home means most are willing to be on less pay."
Reporting by Brenda Goh in SHANGHAI and Jamie Freed in SINGAPORE, with additional reporting by Venus Wu and Anne Marie Roantree in HONG KONG; Editing by Ian Geoghegan
Qingdao Airlines Now Recruiting Foreign A320 Captains
SHENZHEN—In cooperation with Qingdao Airlines, WASINC International is now recruiting A320 captains for this growing, Shandong Province-based airline.
Published Date: May/20/2016
SHENZHEN—In cooperation with Qingdao Airlines, WASINC International is now recruiting A320 captains for this growing, Shandong Province-based airline. Qingdao Airlines operated its first flight in April 2014 and has been growing ever since. With the mission of serving as a boutique carrier for the seaside city of Qingdao, the airline currently operates a fleet of ten Airbus A320s on eight domestic routes from Liuting International Airport in Qingdao. The airline is a partnership between Shandong Airlines, the Nanshan Group, and the Qingdao Transport Development Group. The airline has significant growth in mind with plans to operate 60 A320s by the year 2020. The airline's success since it commenced operations and its plans for growth represent a great opportunity for pilots seeking to fly in China. Pilots are based at the airline's hub in the mountainous coastal city of Qingdao popular with tourists for its temperate climate, numerous parks, and hiking and swimming opportunities. Further information about available positions and benefits can be found on the Jobs tab of WASINC.aero
WASINC Captain Named "Most Advanced" by Shenzhen Airlines
SHENZHEN – WASINC pilot Gregory P. Bujna was recently named Shenzhen Airlines' Most Advanced Employee of 2015.
SHENZHEN – WASINC pilot Gregory P. Bujna was recently named Shenzhen Airlines' Most Advanced Employee of 2015. This award was given in recognition of Captain Bujna's hard work and dedication to the company since joining Shenzhen Airlines in 2008. Bujna expressed his humble appreciation of this honor and thanked the team at WASINC International and his colleagues at Shenzhen Airlines for their support throughout his time working in China. Captain Bujna is the first American pilot to receive this honor from Shenzhen Airlines. WASINC International is committed to providing the best pilots to its airline customers and pilots like Gregory Bujna exemplify the qualities of the ideal expatriate captain.
Ruili Airlines Begins Recruiting Foreign Captains
SHENZHEN – WASINC International has added yet another airline partner to its portfolio of pilot job opportunities in China.
SHENZHEN – WASINC International has added yet another airline partner to its portfolio of pilot job opportunities in China. WASINC has entered into an agreement to recruit and provide pilots for Ruili Airlines of Yunnan Province. The airline was established in 2014 and now flies a fleet of seven Boeing 737NG aircraft on nine routes from its hub in Kunming. The airline is poised to grow signifigantly having signed a commitment to purchase and lease 60 Boeing 737MAX once that aircraft becomes available. This represents a great opportunity for job seekers to join a quickly growing airline in a very historic city boasting one of the best climates in China. WASINC and Ruili Airlines encourage type-rated B737 captains to apply for a position with the company and enjoy the excellent compensation and benefits currently on offer. Further information is available on the Jobs tab of WASINC.aero
GX Airlines Begins Recruiting Foreign Captains
SHENZHEN -- WASINC International is pleased to announce the newest addition to its collection of airline partners and another great job opportunity in China for foreign pilots
Published Date: Mar/30/2016
SHENZHEN -- WASINC International is pleased to announce the newest addition to its collection of airline partners and another great job opportunity in China for foreign pilots. WASINC is now recruiting Embraer E-190 and A320 captains for GX Airlines. Founded in February 2015, GX Airlines is one of China's newest air carriers and represents an exciting opportunity to grow with this budding company. Based in Nanning, Guangxi and operating an expanding fleet currently consisting of eight Embraer E-190s, pilots for GX Airlines have the opportunity to experience life in one of China's Green Cities, rich in tropical foliage and fresh air. The carrier, a joint venture between HNA Group's Tianjin Airlines and the Guanxi Beibu Gulf Investment Group, has a route network currently including nineteen destinations across China with more to come in the near future. The company's impressive plans for expansion combined with their highly appealing compensation packages for foreign captains make GX Airlines a very attractive option for job seekers.
Chengdu Airlines Interviews in Moscow!
Chengdu Airlines will interview A320 Captains in Moscow in May, 2016.
Published Date: Feb/29/2016
SHENZHEN – In cooperation with Chengdu Airlines, WASINC International will host a Captains' Job Interview Fair in Moscow, Russia on May 10-12 and May 14-15, 2016. All are welcome to attend a company presentation and learn more about flying opportunities with Chengdu Airlines. Pilots who have applied and have been invited will have the opportunity to complete face-to-face interviews with Chengdu Airlines and WASINC representatives and to pass their A320 simulator evaluation. The event will be held and accommodations provided at the S7 Airlines facility near Domodedovo Airport just outside Moscow. This is a first of its kind event in Russia for both WASINC and Chengdu Airlines which will be the first step in welcoming many Russian pilots to China to fly with Chengdu Airlines. Chengdu Airlines boasts the highest applicant success rate for foreign pilots of any airline in China, offers competitive compensation packages, and is based in the capital of beautiful Sichuan Province. Further information about the Job Interview Fair and pilot applications can be found on www.cdapilotjobs.com:
WASINC Intl. will host a job fair and interview event for Chengdu Airlines in Moscow.
Dates: May 10,11,12,14,15
Location: S7 Training facility near DME airport, Moscow
For addtional details, and to reserve a slot, visit www.cdapilotjobs.com.
OK Airways New contract $265k/year B737efis/NG
OK Airways has announced dramatic improvements in their compensation.
Published Date: Jan/18/2015
They are hiring B737efis/NG Captains for immediate openings.
Earn up to $265,200 per year.
Please see the details at www.wasinc.aero/jobs.
email us if you are interested at jobs@wasinc.net.
Sichuan Airlines - New Contract!
Sichuan Airlines has announced drastic changes to their compensation, and improved their commuting options
Sichuan Airlines has announced drastic changes to their compensation, and improved their commuting options:
Option A: (45 days of paid leave) $22,950 per month
Option B: (103 days of paid leave) $18,200 per month
Option C: (161 days of paid leave) $14,150 per month
Option D: (4 weeks on, 4 weeks off): $13,050 per month
For more information, please go to www.wasinc.aero/jobs, or send your CV to Xavier Manjarrez at xavier.manjarrez@wasinc.net.
Capital Airlines Job Fair! Miami Nov.10-15
Capital Airlines will be in Miami doing informational briefings about their company as well as interviews for qualified individuals.
Come visit us and take the opportunity to talk one-on-one with executives and Flight Operations managers from China's highest-paying airline.
Dates: Nov. 10-15
Time: Starting at 1300 daily
Who can attend? The briefings are open to everyone.
Who will be interviewed: Current and qualified A320 Captains. They are also interested in talking to Senior First Officers for a possible transition program.
Wasinc International will also be on site to review your resume and talk about your options.
Let us know if you will be attending by contacting us at jobs@wasinc.net.
Capital Airlines pays up to $290/year!
China Lures U.S. Pilots Tired of 14-Year Wait for
China Lures U.S. Pilots Tired of 14-Year Wait for Airline Captain's Seat
Published Date: Apr/11/2014
By Jasmine Wang and Simone Baribeau - 2012
Kent John Krizman has spent 13 years as a co-pilot at American Airlines. For a chance to move across the cockpit, he's ready to take a job in China.
"I should be flying as a captain," said the 52-year-old San Francisco resident, who has 20,000 hours' experience in jet planes. Promotion won't happen for at least five more years at American, while in China it could occur straightaway, he said. He and his wife "are all set to go," he said.
Attendees speak with Sichuan Airlines Co. Ltd. representatives during a pilot recruitment fair in Miami, Florida.
Attendees speak with Sichuan Airlines Co. Ltd. representatives during a pilot recruitment fair in Miami, Florida. Photographer: Joshua Prezant/Bloomberg
Pilot Peter Forbes, right, stands while being interviewed by Shenzhen Airlines employees Hong Guang Liang, left, and Darren Chen during a during a pilot recruitment fair in Miami, Florida.
Pilot Peter Forbes, right, stands while being interviewed by Shenzhen Airlines employees Hong Guang Liang, left, and Darren Chen during a during a pilot recruitment fair in Miami, Florida. Photographer: Joshua Prezant/Bloomberg
Attendees listen to Daniel Lee Fadling, a captain with Hainan Airlines Co. Ltd., unseen, during a pilot recruitment fair in Miami, Florida.
Attendees listen to Daniel Lee Fadling, a captain with Hainan Airlines Co. Ltd., unseen, during a pilot recruitment fair in Miami, Florida. Photographer: Joshua Prezant/Bloomberg
To help lure overseas crew members, Spring Air pays foreign pilots 30 percent more than domestic staff. Photo: China Photos/Getty Images
Attendees head to a conference room to speak with airline representatives during a pilot recruitment fair in Miami, Florida.
Attendees head to a conference room to speak with airline representatives during a pilot recruitment fair in Miami, Florida. Photographer: Joshua Prezant/Bloomberg
Krizman was one of about 550 pilots who attended a China job fair in Miami last week, as first officers find fewer chances for promotion in the U.S. because of slower airline growth and captains retiring later. There are jobs available in China, where a surging economy and a fleet expected to grow 11 percent a year through 2015, according to government forecasts, is creating a need for experienced crewmembers.
"Everyone is facing a pilot shortage," said Shen Wei, head of pilot recruitment at Shanghai-based budget carrier Spring Airlines "Foreign pilots are the quickest option."
To help lure overseas crew members, Spring Air pays foreign pilots 30 percent more than domestic staff, Shen said, without elaboration.
Air China Ltd. , the nation's largest international carrier, was offering $198,000 a year net plus bonuses for Airbus SAS A330 pilots, according to an advertisement on the website of Wasinc International, the recruitment company that helped run the job fair. During the two-day Miami event, which featured about a dozen Chinese airlines, about 70 pilots got provisional job offers, said Scott Snow, a spokesman.
Doubled Pay
Roger Grant, an American Airlines co-pilot, said in Miami that he may be able to about double his salary by moving to China and becoming a captain. He also said a move may offer better long-term prospects.
"I've been worried about the direction that the pilot career has been taking," said the 45-year-old, who lives in Boynton Beach, Florida, with his wife and 7-year-old daughter. Workers across the industry are "getting punished" for mistakes made by major airlines, he said.
It's easier for first officers to become captains in China than the U.S. because of demand rather than lower requirements, said Li Yanhua, an associate professor at Tianjin-based Civil Aviation University of China. Air-traffic controllers in China are already required to speak English, in line with global standards.
China Demand
Nationwide, the number of pilots in China needs to rise to 40,000 from 24,000 in the five years ending 2015, according to a statement posted on the website of the Civil Aviation Administration of China. There are about 1,700 foreign pilots working in the country, according to Spring Air's Shen. Calls to the CAAC went unanswered.
China Southern Airlines Co. , the nation's biggest carrier, is looking to hire 725 pilots this year, including 100 from overseas, it said by e-mail. It employs 4,400 pilots. Air China intends to recruit 600 pilots this year, including as many foreigners as possible, it said. The Beijing-based airline has 46 foreign pilots, or less than 2 percent of its roster.
In the U.S., first officers are finding it more difficult to get promotions as an increase in the mandatoryretirement age for captains to 65 from 60 creates a logjam at the top of chain, said Kit Darby, who runs a pilot-hiring and compensation consulting firm in Peachtree City, Georgia.
Pilots who have been promoted at major U.S. carriers are unlikely to leave as even junior captains earn $12,700 per month on average, plus benefits such as pensions that can boost the package by 40 percent, he said. Moving to China may appeal to the 4 percent of the country's 90,000 pilots that are on furloughs, he said.
"To the furloughed or unemployed pilot an overseas job looks pretty good," he said.
Regional Carriers
Pilots at U.S. regional carriers, which fly smaller planes on short-haul routes, have also been caught by the retirement slowdown as they lose opportunities to move to better-paid positions flying larger models at a major airline.
Tony Giraldo, 51, for instance, said he has spent 15 years flying "numerous hours on the same equipment with no chance for an upgrade" at American Eagle, which ferries passengers from smaller cities to American Airlines' airport hubs. He was considering a move to China as it offers "bigger aircraft and new possibilities," he said.
Some American Airlines pilots recently were promoted to captain, 14 years after being hired, the carrier said. The wait for advancement was five years in the growth period of the 1980s and as long as two decades a few years ago, said Sam Mayer, spokesman for the Allied Pilots Association union.
AMR Bankruptcy
The November bankruptcy filing by AMR Corp., the Fort Worth, Texas-based parent of American Airlines and American Eagle, also spurred Giraldo to consider opportunities elsewhere, he said. Krizman, the American co-pilot, similarly said that concerns about Chapter 11 had "refocused my efforts" to look overseas.
American, which has a hub in Miami, wants to cut 400 pilot jobs as part of bankruptcy restructuring, as well as terminating pensions and outsourcing more flying to other carriers.
The carrier's pilots "will remain highly compensated "even after the proposed changes, said Bruce Hicks, a company spokesman. American crew members "have long been among the best compensated in the industry," he said.
China is stepping up pilot training to help meet demand. The Civil Aviation Flight University of China, the country's biggest training provider, plans to accept 2,400 cadets this year, 33 percent more than last year, it said in e-mailed reply to questions.
Using domestic pilots is simpler for Chinese airlines as there are some restrictions on foreigners flying domestic services, largely because the military controls much of the airspace, said Spring Air's Shen.
"The boom in foreign pilots coming to China may only last a few years," he said. "When we have more choice in the future, I will prefer our own pilots."
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redpajama_set_name: RedPajamaCommonCrawl


Beginning on November 1, 2015, the Digital Trust Foundation will fund a new research project titled "Measuring Cyberstalking and Digital Domestic Abuse Across the Life Span." Led by Amanda Lenhart and Michele Ybarra of the Center for Innovative Public Health Research, and including Jennifer Langhinrichsen-Rohling of the Gulf Coast Behavioral Health and Resiliency Center, this Data & Society Research Institute project will conduct a nationally representative landline and mobile phone survey of 3,000 Americans ages 15 years and older to quantify the prevalence of cyberstalking and digital domestic violence, including cyber-exploitation. Recognizing that witnessing abuse can also have a negative impact, the researchers will further investigate the extent to which people witness others' abusive behavior online. The survey will also examine how online privacy behavior may relate to and even protect against online abuse. The study aims to provide a better understanding of how abuse is perpetrated and experienced through technology, particularly across age groups, and will release a publicly available, publicly accessible report detailing its findings.


Grass Carp by King Sailfish Mounts.
Grass Carp Mounts are available online from King Sailfish Mounts. King Sailfish individually crafts finely detailed Fiberglass Fish Mounts by hand, one-at-a-time. Each Fish Reproduction is airbrushed to resemble that species' most vivid and striking color patterns. Initially formed to promote the release of local Atlantic sailfish, King Sailfish continues to be the leader in Marine Conservation--now offering all sizes and species of Freshwater Fish Mounts, including Grass Carp.
Grass Carp Replicas by King Sailfish Mounts....setting The Standard for Excellence.


text:
"""
    SoftLayer.transports
    ~~~~~~~~~~~~~~~~~~~~
    XML-RPC transport layer that uses the requests library.

    :license: MIT, see LICENSE for more details.
"""
# Required imports to not break existing code.

from .debug import DebugTransport
from .fixture import FixtureTransport
from .rest import RestTransport
from .timing import TimingTransport
from .transport import Request
from .transport import SoftLayerListResult as SoftLayerListResult
from .xmlrpc import XmlRpcTransport

# transports.Request does have a lot of instance attributes. :(
# pylint: disable=too-many-instance-attributes

__all__ = [
    'Request',
    'XmlRpcTransport',
    'RestTransport',
    'TimingTransport',
    'DebugTransport',
    'FixtureTransport',
    'SoftLayerListResult'
]

redpajama_set_name: RedPajamaGithub


For the 51st consecutive year, Callaway Gardens# is welcoming families to its annual Summer Family Adventure – a fun-filled and affordable vacation that invites families to enjoy a host of entertaining activities and exciting adventures, or just kick back and relax while the children are entertained.
Summer Family Adventure captivates guests of all ages and interests with expert-led programs that the whole family will enjoy. If you want to indulge in a current passion, such as golf, tennis or fishing, or try your hand at something new such as water skiing or circus activities – we have it all.# Or perhaps you want to participate in an educational workshop to learn about photography, gardening or horticulture, Callaway Gardens" Summer Family Adventure has that too! It"s a vacation like no other.
With seven-day adventures available each week from June 4 through August 6, Callaway Gardens offers the chance to have summer experiences that make memories to last a lifetime for every member of the family. Summer Family Adventure is ideal for families with diverse interests, who enjoy multi-generational vacations or anyone who loves the outdoors!
Day Camp for children and teens ages 3 to 18 each weekday from 9 a.m. to 3 p.m.
NEW Callaway Gardens TreeTop Adventure featuring aerial challenges for all ages and a NEW Farmers Market featuring locally, regionally available food, flowers and more.
The Summer Family Adventure program offers a variety of programs and activities that allow families to customize their ideal vacation and gives each family member an active, enjoyable summer getaway tailored to their own interests.


Internationally renowned guitarist, arranger and musical director John Jorgenson and The John Jorgenson Quintet will celebrate "Gypsy Jazz" at the Hawai'i Theatre on Saturday, June 4 at 7:30 p.m.
Jorgenson and friends are performing as part of the Django Would Go! Music Festival, presented by Hulavillle Events. Hawaii's own Hot Club of Hulaville will open for them.
Led by Grammy-winning guitarist John Jorgenson, The John Jorgenson Quintet creates a unique musical experience that has enthralled audiences around the world and won innumerable accolades and awards. With each member of the group a recognized virtuoso, The Quintet's style has been dubbed "gypsy jazz" after the dynamic string-driven swing created by Django Reinhardt and Stephane Grappelli in 1930's Paris.
Spanning several decades and many musical styles, John Jorgenson first came to national attention in the mid '80s as co-founder of the successful country-rock act The Desert Rose Band.
Many diverse artists were drawn to add Jorgenson's artistry to their own recordings, including Bonnie Raitt, Bob Seger and Barbara Streisand. In the '90s The Hellecasters gave audiences a chance to experience John's fretboard fireworks in an unrestrained venue and the trio's three original CDs remain favorites of guitarists everywhere. Elton John invited Jorgenson to join his band in 1994 for an 18-month tour that stretched into a 6-year stint of touring, recording and TV appearances.
After discovering Django Reinhardt's music in 1979, Jorgenson became known "the US Ambassador of Gypsy Jazz." The John Jorgenson Quintet is the only American act to ever headline the prestigious Django Reinhardt Memorial Festival in France, and has been featured at other "Djangofests" in the US, UK, Germany and Canada.
For the films "Gattica" and "Head in the Clouds" Jorgenson was tapped to recreate Django's music, and in the latter he even appeared onscreen as Django with stars Charlize Theron and Penelope Cruz.
Adding to the international flavor of the Quintet is Dutch bassist Simon Planting, well known in the gypsy-jazz community from his tenure with legendary Belgian guitarist Fapy Lafertin. Filling out the Quintet are phenomenal young jazz violinist Jason Anick, fiery rhythm guitarist Doug Martin, and veteran percussionist Rick Reed.
The combination of these five incredible musicians creates a sound that has been described as dynamic, romantic, driving, and unforgettable. Legends like Elton John, Earl Scruggs and Duane Eddy along with fellow guitarists such as Tommy Emmanuel, Peter Frampton and Brad Paisley all sing Jorgenson's praises.
Also on the program is Hawaii's top band "The Hot Club of Hulaville" featuring Duane Padilla on violin, Emmet Mahoney on guitar, Ricardo Diaz on contrabass, Sonny Silva on guitar and Ginai as chanteuse.
Filling out the card will be musical guests and a special appearance by guitar great Paul Mehling of the Hot Club of San Francisco and Dan Hicks and His Hot Licks. Plus, audience members shouldn't be surprised if a special guest or two joins in the fun.
Tickets are available online at www.HawaiiTheatre.com or by phone at (808) 528-0506. Prices are $20, $25, $30 and $35, with a special "Gypsy Caravan" premium package at $75.00. Gypsies in the caravan receive premium seating and a meet-the-artists reception with champagne, strawberries and cream, and crepes – both sweet and savory.
Guitar, violin and bass, workshops and master classes with the performers Saturday, June 4 from 10:30 am to 1 pm. For more information, please call (808) 393.6393.
A special show on Maui, 6 pm Sunday, June 5 at the Maui Arts and Cultural Center. There will be workshops and master classes as well beginning at the MACC at 3 pm and a free Gypsy Jam Session at 4 pm. Tickets and information at the MACC Box Office; 242-SHOW (7469) or at www.MauiArts.org.
For additional information on any aspect of the Django Would Go Festival, please call (808) 393-6393.


text:
River Feshie Research
The River Feshie Catchment
Posters, Papers, & More
Utah State University - Watershed Sciences
USU ET-AL Lab
Queen Mary University - Geography
University of Southampton - Geography
University of Leeds - Geography
Forest Sciences Centre of Catalonia
Home > Our Research >
Geomorphic Change Detection
Repeat surveys of streams can be immensely insightful in terms of understanding how channel form and associated habitat changes over the course of a single flood, or over annual or longer timescales. The River Feshie has been an instrumental site for the development and testing of tools used for geomorphic change detection (GCD). The procedure behind GCD is quite simple, and at its core involves a three-step process:
Conduct a survey of a landscape and produce a digital representation of that landscape (called a digital elevation model, or DEM).
Return to the site some time later, perhaps after a flood, or perhaps a year (or more) later; conduct another survey and produce a second DEM from the data.
Subtract the old DEM from the new DEM to produce a map of changes that occurred on the landscape. This resultant dataset is termed a "DEM-of-Difference" or DoD.
In visual terms, the production of a DoD is simply the result of subtracting an older elevation surface from a newer one:
Given the multi-year survey record which exists for the River Feshie, and because of its dynamic nature as a braided stream, it provides an ideal test case for producing DoDs. Here we show a series of DoDs produced from surveys conducted annually between 2003 and 2007.
DoDs produced using DEMs from 2003 to 2007 on the River Feshie. The DEMs (top row) have been detrended (that is, they have had the down-valley slope component of elevation change removed) prior to DoD calculation. DoDs are shown in the bottom row, with areas undergoing erosion in red and those undergoing deposition in blue. Figure is from Wheaton et al. (2010), copyright Wiley and Sons.
While DoDs alone can be useful for examining channel changes over a variety of timescales, we must also consider the reliability of the data from which they are produced. That is, how much of the change that we see in the resulting DoD is the result of actual geomorphic shifts in the elevation of the landscape, versus how much of the change is due to inherent error or uncertainty (termed 'noise') in the surveys from which the DEMs (and eventually DoDs) are computed?
In a 2010 paper, members of our research team (led by Joe Wheaton) examined the influence of survey noise on DoDs. The figure below shows that depending on the reliability of the data from which DEMs are built, the amount of elevation change that we are able to detect in resultant DoDs can vary tremendously. In each DoD below, the 'minimum level of detection (minLoD)' - that is, the smallest elevation differences that can our instruments can reliably survey - is varied hypothetically, and the ultimate result on the DoD is shown. Note that with a higher minLoD (that is, hypothetically less reliable vertical resolution of DEMs), the volume of change that we see in the corresponding DoD is altered.
The significance of minLOD on resultant DoDs and associated volumetric change estimates. DoD maps are shown on top and elevation change distributions (which are histograms of volume of change assigned to a given elevation change bin) are shown on bottom. The gross unthresholded DoD is shown on the far left, and moving toward the right progressively more conservative DoDs (i.e. higher minLoD) are shown. Figure is from Wheaton et al. (2010), copyright Wiley and Sons.
Knowing that the changes we can resolve in our DoDs are ultimately a function of the reliability of our topographic surveys from which DEMs are built, how can we incorporate survey uncertainty into the calculation of a DoD? There are several measurable quantities we can use to do this. In Wheaton et al. (2010), a combination of point density (higher density leads to a more accurate DEM), surface slope (steeper sloping areas result in higher uncertainty as they are more difficult to capture in surveys, leading to lower-quality DEMs), and GPS quality (lower point quality, as recorded by the instrument, leads to lower DEM quality) were used as inputs to a fuzzy inference system, and the combination of these three values yielded an estimate of DEM reliability at each point on these elevation surfaces. An example of the inputs to this fuzzy inference system are shown below.
Illustration of fuzzy inference system (FIS) construction and resulting probability map for 2006-2005 DoD. Inputs to the FIS include (a) survey point density, (b) surface slope, and (c) GPS point quality. Resulting map of elevation uncertainty for use in DoD calculation is shown at far right. Figure is from Wheaton et al. (2010), copyright Wiley and Sons.
When applied to the DEMs from the River Feshie, the use of the fuzzy inference system allows for removal of inherent survey noise by using measurable quantities derived from survey data. The result is a 'cleaner' DoD that in effect allows for much greater confidence that the geomorphic change we see is 'real' and not the result of survey errors.
Comparison of thresholded DoDs (at 95% confidence that changes seen are real and not due to survey noise) based on applying fuzzy inference system (FIS) are shown on top. Bottom row of DoDs also includes the application of a spatial coherence filter; that is, changes are more likely to be real (and not an artifact of survey noise) if they are surrounded, or spatially coherent, with other geomorphic change in a similar direction. Figure is from Wheaton et al. (2010), copyright Wiley and Sons.
The principles of geomorphic change detection developed on the Feshie have been instrumental in the development of software designed to allow users to produce DoDs using their own survey data. The Geomorphic Change Detection software is freely available here. You can find more information about the theory behind GCD in:
Wheaton JM, Brasington J, Darby SE, Sear DA. 2010. Accounting for uncertainty in DEMs from repeate topographic surveys: improved sediment budgets. Earth Surface Processes and Landforms 35: 136-156. DOI: 10.1002/esp.1886
Wheaton JM, Brasington J, Darby SE, Merz JE, Pasternack GB, Sear DA and Vericat D. 2010. Linking Geomorphic Changes to Salmonid Habitat at a Scale Relevant to Fish. River Research and Applications 26: 469-486. DOI: 10.1002/rra.1305.
Wheaton JM. 2008. Uncertainty in Morphological Sediment Budgeting of Rivers. Unpublished PhD Thesis, University of Southampton, Southampton, 412 pp.
River Feshie Website & Materials by USU ET-AL (2011-2013) is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.


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For decades skaters have been awestruck by a piece of wood with wheels and how they could perform tricks that seem to defy gravity. Skateboarding was created by surfers for when there were no waves and is continuing to grow as an action sport with a growing number of participants. As one of those skaters I love the feeling of freedom you get when are skating. It is how i prefer to get to places. Skating is not a hobby it is a lifestyle. Skating is also primarily physics when it comes to performing tricks, but this is the chemistry of a skateboard.Composition of ...The deck varies but is usually 50% Carbon, 42% Oxygen, 6% Hydrogen, 1% Nitrogen and 1% of other elements.
The trucks are made of aluminum (Al)
The wheels are made of polyurethane (C3H8N2O)
Every part of the skateboard is needed for it to work properly. Like pro skater Chris Cole said "It's a strange phenomenon how this piece of wood, wheels and a turning system has made so many people so happy."
The deck is made of 7 plys of wood that is usually maple wood (an alternative is bamboo). It is cut to have a width of 7-10 in. and length is 28 to 32 in. The cut makes sure that the nose is steeper than the tail and has a dip in the middle which is known as the concave. It is then glued together and has heat printed graphic on the bottom side. If the board is not constructed properly it would snap quickly after an impact.
The Trucks are made of aluminum and has small rubber bushings that are pushed down on by a nut called the king pin. This allows for control of turning.
The wheels are made of polyurethane which is hard enough for streets and doesn't wear away quickly. The size of the wheel is also important for speed. Big wheels can go faster and go over cracks easier but are less responsive. Smaller wheels are used for tricks because they are lighter and more responsive.
The bearings are made of Silicon Nitride or ceramic ball bearings that consists of 7 small balls that give the wheel constant movement. There are 2 bearings for each wheel.
The griptape is made of either Silicon Carbide which is naturally black in color or made of Aluminum Oxide and clear in color which allows for a design underneath. It is like sandpaper but allows your feet to stay on the board without slipping.
https://en.wikipedia.org/wiki/Skateboard#Wheels
Info on the board and its history
https://en.wikipedia.org/wiki/Polyurethane
What makes polyurethane
http://www.chemnet.com/cas/en/9009-54-5/polyurethane-foam.html
More on polyurethane
https://www.warehouseskateboards.com/help/Skateboard-Decks-Buying-Guide
More info on the skateboard
https://www.warehouseskateboards.com/blog/2008/09/26/skateboard-trucks-what-size-do-you-need/
Info on the trucks
http://www.tactics.com/info/choosing-skateboard-wheels
Info on the wheel
http://www.ratvision.com/index.php/91-article-mosiacs/info-article-mosiacs/grip-tape/30-what-grip-tape-is-better
Info on griptape
http://www.guidechem.com/cas-120/12033-89-5.html
http://dwayneportfolio.weebly.com/teaching-blog/aluminum-oxide-al2o3-lewis-diagramstructuremodel
Simon Hanson is a junior at Billings Senior High School (Class of 2017) and plays soccer for the Magic City S.C. U18 team. He loves music and produces some of his own music in the electronic Dance Music genre. He also enjoys Mixed Martial Arts and skateboarding. He interested in becoming a heart surgeon or DJ on the professional circuit around the world. His family is also a member of the Scottish clan Malcolm.


"Hello. Looking for overflow work in OC/Long Beach area, Ontario/I.E. area in work comp. Please feel free to call or text. Thank you, Lynette Guzman, CSR 12091 562-685-2063"
Hello. Looking for overflow work in OC/Long Beach area, Ontario/I.E. area in work comp. Please feel free to call or text.


Why do we get used to smells?
Your nose might be hiding things from you about familiar smells.
Our nervous system has evolved to become progressively less sensitive to a stimulus, the longer it persists. This enables us to concentrate on the newest sensations that are more likely to be an opportunity or a threat. We also have an olfactory memory that discards smells that we have experienced recently. This means that you don't notice the smell of your house when you come home from work, but it smells strange when you come back from holiday.