Calculus Examples

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

Step 1

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

Simplify the numerator.

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Step 6.1.1.1

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

$\frac{(x + 4) (x - 4)}{4 x - 16}$

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Factor $4$ out of $4 x - 16$ .

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Step 6.1.2.1.1

Factor $4$ out of $4 x$ .

$\frac{(x + 4) (x - 4)}{4 (x) - 16}$

Step 6.1.2.1.2

Factor $4$ out of $- 16$ .

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

Step 6.1.2.1.3

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

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

Step 6.1.2.2

Cancel the common factor of $x - 4$ .

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Step 6.1.2.2.1

Cancel the common factor.

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

Step 6.1.2.2.2

Rewrite the expression.

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

Step 6.2

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$4$

$x$

$4$

Step 6.3

Divide the highest order term in the dividend $x$ by the highest order term in divisor $4$ .

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

Step 6.4

Multiply the new quotient term by the divisor.

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

Step 6.5

The expression needs to be subtracted from the dividend, so change all the signs in $x$

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

Step 6.6

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

		$\frac{x}{4}$
$4$		$x$	+	$4$
	-	$x$
		$0$

Step 6.7

Pull the next terms from the original dividend down into the current dividend.

		$\frac{x}{4}$
$4$		$x$	+	$4$
	-	$x$
		$0$	+	$4$

Step 6.8

The final answer is the quotient plus the remainder over the divisor.

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

Step 6.9

Split the solution into the polynomial portion and the remainder.

$\frac{x}{4} + 1$

Step 6.10

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

$y = \frac{x}{4}$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{x}{4}$

Step 8