

# Limits and continuity

We begin our study of analysis with limits and continuity. These concepts are fundamental to understanding the behavior of functions and their derivatives.

A function is said to be continuous at a point $a$ if the limit of the function as $x$ approaches $a$ exists. Mathematically, this can be written as:

$$\lim_{x \to a} f(x) = f(a)$$

Continuity is an important property of functions because it allows us to make meaningful statements about the behavior of a function near a point. For example, if a function is continuous at a point, we can say that the function has a well-defined value at that point.

Consider the function $f(x) = \frac{1}{x}$. The function is continuous at $x = 0$ because the limit of the function as $x$ approaches $0$ exists:

$$\lim_{x \to 0} \frac{1}{x} = \infty$$

However, the function is not continuous at any other point because the limit does not exist.

# Derivatives: definitions and basic rules



The derivative of a function gives us the rate of change of the function with respect to its input. It is defined as the limit of the difference quotient:

$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

There are several basic rules for working with derivatives:

- The derivative of a constant is $0$.
- The derivative of a sum of functions is the sum of the derivatives of the individual functions.
- The derivative of a product of functions is the product of the derivatives of the individual functions.

# The product and quotient rules



The product rule states that if $u(x)$ and $v(x)$ are functions, then the derivative of their product is:

$$(u(x)v(x))' = u'(x)v(x) + u(x)v'(x)$$

The quotient rule states that if $u(x)$ and $v(x)$ are functions, then the derivative of their quotient is:

$$\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$$

# The chain rule



The chain rule states that if $u(x) = f(g(x))$, then the derivative of $u(x)$ with respect to $x$ is:

$$u'(x) = f'(g(x))g'(x)$$

# The power rule



The power rule states that if $u(x) = f(x)^n$, where $n$ is a constant, then the derivative of $u(x)$ with respect to $x$ is:

$$u'(x) = nf(x)^{n-1}f'(x)$$

# The sum rule



The sum rule states that if $u(x) = f(x) + g(x)$, then the derivative of $u(x)$ with respect to $x$ is:

$$u'(x) = f'(x) + g'(x)$$

# Integration: definite and indefinite



Integration is the inverse process of differentiation. It allows us to find the area under a curve, or the accumulated value of a function over a specified interval.

There are two types of integration: definite and indefinite. Definite integration finds the area under a curve between two specified points, while indefinite integration finds the accumulated value of a function over a specified interval.

The fundamental theorem of calculus connects the two types of integration. It states that the definite integral of a function is equal to the difference between the accumulated value of the function at two points.

# The fundamental theorem of calculus



The fundamental theorem of calculus states that if $f(x)$ is a continuous function on the interval $[a, b]$, then:

$$\int_a^b f(x)dx = F(b) - F(a)$$

where $F(x)$ is the indefinite integral of $f(x)$.

# Sequences and series



A sequence is an ordered list of numbers. A series is a sum of terms in a sequence.

Convergence refers to the behavior of a sequence or series as the terms become infinitely large. There are several types of convergence: absolute convergence, conditional convergence, and divergence.

# Convergence and divergence



A sequence or series is said to converge if the terms of the sequence or series approach a specific limit as the terms become infinitely large.

A sequence or series is said to diverge if the terms of the sequence or series do not approach a specific limit as the terms become infinitely large.

# The p-series and integral test



The p-series test is used to determine whether a series converges or diverges based on the value of $p$. The integral test is used to determine whether a series converges or diverges based on the behavior of its integral.

Both tests are important tools for studying the convergence and divergence of series.