Precalculus Examples

$f (x) = \frac{x^{2} - 9}{x - 3}$

Step 1

Find where the expression $\frac{x^{2} - 9}{x - 3}$ is undefined.

$x = 3$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

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 $m$ .

$n = 2$

$m = 1$

Step 5

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Simplify the expression.

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Step 6.1.1

Simplify the numerator.

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Step 6.1.1.1

Rewrite $9$ as $3^{2}$ .

$\frac{x^{2} - 3^{2}}{x - 3}$

Step 6.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{(x + 3) (x - 3)}{x - 3}$

Step 6.1.2

Cancel the common factor of $x - 3$ .

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Step 6.1.2.1

Cancel the common factor.

$\frac{(x + 3) (x - 3)}{x - 3}$

Step 6.1.2.2

Divide $x + 3$ by $1$ .

$x + 3$

Step 6.2

The oblique asymptote is the polynomial portion of the long division result.

$y = x + 3$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = x + 3$

Step 8