Precalculus Examples

$y = \frac{4 x^{2} - 16}{x - 2}$

Step 1

Find where the expression $\frac{4 x^{2} - 16}{x - 2}$ 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

Factor $4$ out of $4 x^{2} - 16$ .

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Step 6.1.1.1.1

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

$\frac{4 (x^{2}) - 16}{x - 2}$

Step 6.1.1.1.2

Factor $4$ out of $- 16$ .

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

Step 6.1.1.1.3

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

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

Step 6.1.1.2

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

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

Step 6.1.1.3

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 $b = 2$ .

$\frac{4 (x + 2) (x - 2)}{x - 2}$

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Cancel the common factor of $x - 2$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{4 (x + 2) (x - 2)}{x - 2}$

Step 6.1.2.1.2

Divide $4 (x + 2)$ by $1$ .

$4 (x + 2)$

Step 6.1.2.2

Apply the distributive property.

$4 x + 4 \cdot 2$

Step 6.1.2.3

Multiply $4$ by $2$ .

$4 x + 8$

Step 6.2

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

$y = 4 x + 8$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = 4 x + 8$

Step 8