Example 6. Find the asymptotes of the curve $y=\frac{x^{2}}{\sqrt{x^{2}-1}}$.
Solution. By equating the denominator to zero, we obtain two vertical asymptotes: $x=-1, x=1$.

Using formulas (3.13) and (3.14), we find the oblique asymptotes.

We have:

$$
\begin{gathered}
k=\lim _{x \rightarrow+\infty} \frac{y}{x}=\lim _{x \rightarrow+\infty} \frac{x^{2}}{x \sqrt{x^{2}-1}}=1 \\
b=\lim _{x \rightarrow+\infty}(y-x)=\lim _{x \rightarrow+\infty} \frac{x^{2}-x \sqrt{x^{2}-1}}{\sqrt{x^{2}-1}}=0
\end{gathered}
$$

Thus, the line $y=x$ is the right oblique asymptote. Further,

$$
k_{1}=\lim _{x \rightarrow-\infty} \frac{y}{x}=-1, b_{1}=\lim _{x \rightarrow-\infty}(y-x)=0
$$

so $y=-x$ is also an asymptote (Fig. 3.17).

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-131.jpg?height=403&width=505&top_left_y=67&top_left_x=41)

Fig. 3.17

The second oblique asymptote could also be obtained based on the symmetry of the curve with respect to the $Oy$ axis.

## Problems

Find the intervals of concavity and inflection points of the graphs of the functions:

1. $y=\ln x$.
2. $y=x^{3}-6 x^{2}+9 x+5$.
3. $y=\frac{3}{8} x^{4}-x^{3}+2$.
4. $y=x^{3}+\frac{x^{4}}{4}$.

Find the asymptotes of the curves:
5. $y=\frac{2}{x+3}$.
6. $y=\frac{6}{x^{2}-16}$.
7. $y=\frac{x^{2}+8 x-6}{x}$.
8. $y=\frac{x^{2}}{x+4}$.
9. $y=e^{\frac{1}{x}}$.
10. $b^{2} x^{2}+a^{2} y^{2}=x^{2} y^{2}$.

## Answers

1. The curve is concave downward. 2. In the interval $(-\infty, 2)$ the curve is concave downward, in the interval $(2, \infty)$ it is concave upward, $M(2,7)$ is the inflection point. 3. $M(0,2)$, $N\left(\frac{4}{3}, \frac{22}{27}\right)$ are inflection points. 4. $M(0,0)$, $N(-2,-4)$. 5. $x=-3$. 6. $x=-4$, $x=4$. 7. $x=0$, $y=x+8$. 8. $x=-4$, $y=x-4$. 9. $y=1$. 10. $x= \pm a$, $y= \pm b$.

## § 3.5. Investigation of Functions and Construction of Their Graphs

The investigation of functions can be carried out according to the following scheme:

1. Find the domain of the function.
2. Investigate the behavior of the function as $x$ approaches the endpoints of the domain intervals.
3. Find the intervals of increase and decrease of the function, and the points of extremum.
4. Calculate the values of the extrema.
5. Determine the intervals of convexity and concavity of the graph, and find the inflection points.
6. Find the points of intersection of the graph of the function with the coordinate axes.
7. Find the asymptotes of the graph of the function.

Based on the results of the investigation, a mathematically accurate sketch of its graph can be constructed.

If the function is even or odd, it is sufficient to investigate the function and construct its graph for positive values of the argument from the domain.

Sometimes the order of investigation can be chosen based on the specific features of the given function.