Precalculus Examples

$f (x) = \frac{9 x^{2} - 25}{9 x^{2} + 78 x + 105}$

Step 1

Find where the expression $\frac{9 x^{2} - 25}{9 x^{2} + 78 x + 105}$ is undefined.

$x = - 7, x = - \frac{5}{3}$

Step 2

Since $\frac{9 x^{2} - 25}{9 x^{2} + 78 x + 105}$ $\to$ $\infty$ as $x$ $\to$ $- 7$ from the left and $\frac{9 x^{2} - 25}{9 x^{2} + 78 x + 105}$ $\to$ $- \infty$ as $x$ $\to$ $- 7$ from the right, then $x = - 7$ is a vertical asymptote.

$x = - 7$

Step 3

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 4

Find $n$ and $m$ .

$n = 2$

$m = 2$

Step 5

Since $n = m$ , the horizontal asymptote is the line $y = \frac{a}{b}$ where $a = 9$ and $b = 9$ .

$y = 1$

Step 6

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 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 7$

Horizontal Asymptotes: $y = 1$

No Oblique Asymptotes

Step 8