Precalculus Examples

$f (x) = \frac{1}{x^{2} - 4}$

Step 1

Find where the expression

$\frac{1}{x^{2} - 4}$ is undefined.

$x = - 2, x = 2$

Step 2

Since

$\frac{1}{x^{2} - 4}$

$\to$

$\infty$ as

$x$

$\to$

$- 2$ from the left and

$\frac{1}{x^{2} - 4}$

$\to$

$- \infty$ as

$x$

$\to$

$- 2$ from the right, then

$x = - 2$ is a vertical asymptote.

$x = - 2$

Step 3

Since

$\frac{1}{x^{2} - 4}$

$\to$

$- \infty$ as

$x$

$\to$

$2$ from the left and

$\frac{1}{x^{2} - 4}$

$\to$

$\infty$ as

$x$

$\to$

$2$ from the right, then

$x = 2$ is a vertical asymptote.

$x = 2$

Step 4

List all of the vertical asymptotes:

$x = - 2, 2$

Step 5

Consider the rational function

$R (x) = \frac{a x^{n}}{b x^{m}}$ where

$n$ is the degree of the numerator and

$m$ is the degree of the denominator.

1. If

$n < m$ , then the x-axis,

$y = 0$ , is the horizontal asymptote.

2. If

$n = m$ , then the horizontal asymptote is the line

$y = \frac{a}{b}$ .

3. If

$n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 6

Find

$n$ and

$m$ .

$n = 0$

$m = 2$

Step 7

Since

$n < m$ , the x-axis,

$y = 0$ , is the horizontal asymptote.

$y = 0$

Step 8

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 9

This is the set of all asymptotes.

Vertical Asymptotes:

$x = - 2, 2$

Horizontal Asymptotes:

$y = 0$

No Oblique Asymptotes

Step 10

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