Precalculus Examples

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

Step 1

Find where the expression $\frac{x^{2} - 4}{- x - 2}$ is undefined.

$x = - 2$

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 $4$ as $2^{2}$ .

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

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 $b = 2$ .

$\frac{(x + 2) (x - 2)}{- x - 2}$

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Cancel the common factor of $x + 2$ and $- x - 2$ .

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Step 6.1.2.1.1

Factor $- 1$ out of $x$ .

$\frac{(- 1 (- x) + 2) (x - 2)}{- x - 2}$

Step 6.1.2.1.2

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{(- 1 (- x) - 1 (- 2)) (x - 2)}{- x - 2}$

Step 6.1.2.1.3

Factor $- 1$ out of $- 1 (- x) - 1 (- 2)$ .

$\frac{- 1 (- x - 2) (x - 2)}{- x - 2}$

Step 6.1.2.1.4

Cancel the common factor.

$\frac{- 1 (- x - 2) (x - 2)}{- x - 2}$

Step 6.1.2.1.5

Divide $- 1 (x - 2)$ by $1$ .

$- 1 (x - 2)$

Step 6.1.2.2

Rewrite $- 1 (x - 2)$ as $- (x - 2)$ .

$- (x - 2)$

Step 6.1.2.3

Apply the distributive property.

$- x - - 2$

Step 6.1.2.4

Multiply $- 1$ by $- 2$ .

$- x + 2$

Step 6.2

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

$y = - x + 2$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = - x + 2$

Step 8