Precalculus Examples

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

Step 1

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

$x = - 4$

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

Factor $x$ out of $x^{2} + 4 x$ .

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Step 6.1.1.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x + 4 x}{- x - 4}$

Step 6.1.1.2

Factor $x$ out of $4 x$ .

$\frac{x \cdot x + x \cdot 4}{- x - 4}$

Step 6.1.1.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$\frac{x (x + 4)}{- x - 4}$

Step 6.1.2

Cancel the common factor of $x + 4$ and $- x - 4$ .

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Step 6.1.2.1

Factor $- 1$ out of $x$ .

$\frac{x (- 1 (- x) + 4)}{- x - 4}$

Step 6.1.2.2

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{x (- 1 (- x) - 1 (- 4))}{- x - 4}$

Step 6.1.2.3

Factor $- 1$ out of $- 1 (- x) - 1 (- 4)$ .

$\frac{x (- 1 (- x - 4))}{- x - 4}$

Step 6.1.2.4

Cancel the common factor.

$\frac{x (- 1 (- x - 4))}{- x - 4}$

Step 6.1.2.5

Divide $x \cdot (- 1)$ by $1$ .

$x \cdot (- 1)$

Step 6.1.3

Simplify the expression.

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Step 6.1.3.1

Move $- 1$ to the left of $x$ .

$- 1 x$

Step 6.1.3.2

Rewrite $- 1 x$ as $- x$ .

$- x$

Step 6.2

Since there is no polynomial portion from the polynomial division, there are no oblique asymptotes.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

No Oblique Asymptotes

Step 8