Precalculus Examples

$f (x) = \frac{x^{2}}{\sqrt{x^{2} - 1}}$

Step 1

Find where the expression $\frac{x^{2}}{\sqrt{x^{2} - 1}}$ is undefined.

$- 1 \leq x \leq 1$

Step 2

Since $\frac{x^{2}}{\sqrt{x^{2} - 1}}$ $\to$ $\infty$ as $x$ $\to$ $- 1$ from the left and $\frac{x^{2}}{\sqrt{x^{2} - 1}}$ $\to$ $\infty$ as $x$ $\to$ $- 1$ from the right, then $x = - 1$ is a vertical asymptote.

$x = - 1$

Step 3

Since $\frac{x^{2}}{\sqrt{x^{2} - 1}}$ $\to$ $\infty$ as $x$ $\to$ $1$ from the left and $\frac{x^{2}}{\sqrt{x^{2} - 1}}$ $\to$ $\infty$ as $x$ $\to$ $1$ from the right, then $x = 1$ is a vertical asymptote.

$x = 1$

Step 4

List all of the vertical asymptotes:

$x = - 1, 1$

Step 5

Since the limit does not exist, there are no horizontal asymptotes.

No Horizontal Asymptotes

Step 6

Use polynomial division to find the oblique asymptotes. Because this expression contains a radical, polynomial division cannot be performed.

Cannot Find Oblique Asymptotes

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 1, 1$

No Horizontal Asymptotes

Cannot Find Oblique Asymptotes

Step 8