Calculus Examples

$f (x) = \frac{x^{2} - 36}{2 x + 5}$

Step 1

Find where the expression $\frac{x^{2} - 36}{2 x + 5}$ is undefined.

$x = - \frac{5}{2}$

Step 2

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 3

Find $n$ and $m$ .

$n = 2$

$m = 1$

Step 4

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite $36$ as $6^{2}$ .

$\frac{x^{2} - 6^{2}}{2 x + 5}$

Step 5.1.2

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 $b = 6$ .

$\frac{(x + 6) (x - 6)}{2 x + 5}$

Step 5.2

Expand $(x + 6) (x - 6)$ .

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Step 5.2.1

Apply the distributive property.

$\frac{x (x - 6) + 6 (x - 6)}{2 x + 5}$

Step 5.2.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 6 + 6 (x - 6)}{2 x + 5}$

Step 5.2.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 6 + 6 x + 6 \cdot - 6}{2 x + 5}$

Step 5.2.4

Reorder $x$ and $- 6$ .

$\frac{x \cdot x - 6 x + 6 x + 6 \cdot - 6}{2 x + 5}$

Step 5.2.5

Raise $x$ to the power of $1$ .

$\frac{x \cdot x - 6 x + 6 x + 6 \cdot - 6}{2 x + 5}$

Step 5.2.6

Raise $x$ to the power of $1$ .

$\frac{x \cdot x - 6 x + 6 x + 6 \cdot - 6}{2 x + 5}$

Step 5.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} - 6 x + 6 x + 6 \cdot - 6}{2 x + 5}$

Step 5.2.8

Add $1$ and $1$ .

$\frac{x^{2} - 6 x + 6 x + 6 \cdot - 6}{2 x + 5}$

Step 5.2.9

Multiply $6$ by $- 6$ .

$\frac{x^{2} - 6 x + 6 x - 36}{2 x + 5}$

Step 5.2.10

Add $- 6 x$ and $6 x$ .

$\frac{x^{2} + 0 - 36}{2 x + 5}$

Step 5.2.11

Subtract $36$ from $0$ .

$\frac{x^{2} - 36}{2 x + 5}$

Step 5.3

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

$2 x$

$5$

$x^{2}$

$0 x$

$36$

Step 5.4

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $2 x$ .

					$\frac{x}{2}$
	$2 x$	+	$5$		$x^{2}$	+	$0 x$	-	$36$

Step 5.5

Multiply the new quotient term by the divisor.

				$\frac{x}{2}$
$2 x$	+	$5$		$x^{2}$	+	$0 x$	-	$36$
			+	$x^{2}$	+	$\frac{5 x}{2}$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + \frac{5 x}{2}$

				$\frac{x}{2}$
$2 x$	+	$5$		$x^{2}$	+	$0 x$	-	$36$
			-	$x^{2}$	-	$\frac{5 x}{2}$

Step 5.7

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

				$\frac{x}{2}$
$2 x$	+	$5$		$x^{2}$	+	$0 x$	-	$36$
			-	$x^{2}$	-	$\frac{5 x}{2}$
					-	$\frac{5 x}{2}$

Step 5.8

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

				$\frac{x}{2}$
$2 x$	+	$5$		$x^{2}$	+	$0 x$	-	$36$
			-	$x^{2}$	-	$\frac{5 x}{2}$
					-	$\frac{5 x}{2}$	-	$36$

Step 5.9

Divide the highest order term in the dividend $- \frac{5 x}{2}$ by the highest order term in divisor $2 x$ .

				$\frac{x}{2}$	-	$\frac{5}{4}$
$2 x$	+	$5$		$x^{2}$	+	$0 x$	-	$36$
			-	$x^{2}$	-	$\frac{5 x}{2}$
					-	$\frac{5 x}{2}$	-	$36$

Step 5.10

Multiply the new quotient term by the divisor.

				$\frac{x}{2}$	-	$\frac{5}{4}$
$2 x$	+	$5$		$x^{2}$	+	$0 x$	-	$36$
			-	$x^{2}$	-	$\frac{5 x}{2}$
					-	$\frac{5 x}{2}$	-	$36$
					-	$\frac{5 x}{2}$	-	$\frac{25}{4}$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $- \frac{5 x}{2} - \frac{25}{4}$

				$\frac{x}{2}$	-	$\frac{5}{4}$
$2 x$	+	$5$		$x^{2}$	+	$0 x$	-	$36$
			-	$x^{2}$	-	$\frac{5 x}{2}$
					-	$\frac{5 x}{2}$	-	$36$
					+	$\frac{5 x}{2}$	+	$\frac{25}{4}$

Step 5.12

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

				$\frac{x}{2}$	-	$\frac{5}{4}$
$2 x$	+	$5$		$x^{2}$	+	$0 x$	-	$36$
			-	$x^{2}$	-	$\frac{5 x}{2}$
					-	$\frac{5 x}{2}$	-	$36$
					+	$\frac{5 x}{2}$	+	$\frac{25}{4}$
							-	$\frac{119}{4}$

Step 5.13

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

$\frac{x}{2} - \frac{5}{4} - \frac{119}{4 (2 x + 5)}$

Step 5.14

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

$y = \frac{x}{2} - \frac{5}{4}$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{5}{2}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{x}{2} - \frac{5}{4}$

Step 7