Precalculus Examples

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

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

Step 1

Find where the expression

\frac{x^{2} - 9}{x - 3}

$\frac{x^{2} - 9}{x - 3}$ is undefined.

x = 3

$x = 3$

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}}

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

n

$n$ is the degree of the numerator and

m

$m$ is the degree of the denominator.

1. If

n < m

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

y = 0

$y = 0$ , is the horizontal asymptote.

2. If

n = m

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

y = \frac{a}{b}

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

3. If

n > m

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

Step 4

Find

n

$n$ and

m

$m$ .

n = 2

$n = 2$

m = 1

$m = 1$

Step 5

Since

n > m

$n > m$ , 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

$9$ as

3^{2}

$3^{2}$ .

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

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

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)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 3

$b = 3$ .

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

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

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

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

Step 6.1.2

Cancel the common factor of

x - 3

$x - 3$ .

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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