Calculus Examples

$\frac{- x^{2} - 4 x - 7}{x + 3}$

Step 1

Find where the expression $- \frac{x^{2} + 4 x + 7}{x + 3}$ is undefined.

$x = - 3$

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 expression.

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Step 5.1.1

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) - 4 x - 7}{x + 3}$

Step 5.1.2

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

$\frac{- (x^{2}) - (4 x) - 7}{x + 3}$

Step 5.1.3

Factor $- 1$ out of $- (x^{2}) - (4 x)$ .

$\frac{- (x^{2} + 4 x) - 7}{x + 3}$

Step 5.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{- (x^{2} + 4 x) - 1 (7)}{x + 3}$

Step 5.1.5

Factor $- 1$ out of $- (x^{2} + 4 x) - 1 (7)$ .

$\frac{- (x^{2} + 4 x + 7)}{x + 3}$

Step 5.1.6

Simplify the expression.

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Step 5.1.6.1

Rewrite $- (x^{2} + 4 x + 7)$ as $- 1 (x^{2} + 4 x + 7)$ .

$\frac{- 1 (x^{2} + 4 x + 7)}{x + 3}$

Step 5.1.6.2

Move the negative in front of the fraction.

$- \frac{x^{2} + 4 x + 7}{x + 3}$

Step 5.2

Expand $- (x^{2} + 4 x + 7)$ .

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Step 5.2.1

Negate $x^{2} + 4 x + 7$ .

$\frac{- (x^{2} + 4 x + 7)}{x + 3}$

Step 5.2.2

Apply the distributive property.

$\frac{- (x^{2} + 4 x) - 1 \cdot 7}{x + 3}$

Step 5.2.3

Apply the distributive property.

$\frac{- x^{2} - (4 x) - 1 \cdot 7}{x + 3}$

Step 5.2.4

Remove parentheses.

$\frac{- x^{2} - 1 \cdot (4 x) - 1 \cdot 7}{x + 3}$

Step 5.2.5

Multiply $- 1$ by $4$ .

$\frac{- x^{2} - 4 x - 1 \cdot 7}{x + 3}$

Step 5.2.6

Multiply $- 1$ by $7$ .

$\frac{- x^{2} - 4 x - 7}{x + 3}$

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 $0$ .

$x$

$3$

$x^{2}$

$4 x$

$7$

Step 5.4

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

				-	$x$
	$x$	+	$3$	-	$x^{2}$	-	$4 x$	-	$7$

Step 5.5

Multiply the new quotient term by the divisor.

			-	$x$
$x$	+	$3$	-	$x^{2}$	-	$4 x$	-	$7$
			-	$x^{2}$	-	$3 x$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} - 3 x$

			-	$x$
$x$	+	$3$	-	$x^{2}$	-	$4 x$	-	$7$
			+	$x^{2}$	+	$3 x$

Step 5.7

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

			-	$x$
$x$	+	$3$	-	$x^{2}$	-	$4 x$	-	$7$
			+	$x^{2}$	+	$3 x$
					-	$x$

Step 5.8

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

			-	$x$
$x$	+	$3$	-	$x^{2}$	-	$4 x$	-	$7$
			+	$x^{2}$	+	$3 x$
					-	$x$	-	$7$

Step 5.9

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

			-	$x$	-	$1$
$x$	+	$3$	-	$x^{2}$	-	$4 x$	-	$7$
			+	$x^{2}$	+	$3 x$
					-	$x$	-	$7$

Step 5.10

Multiply the new quotient term by the divisor.

			-	$x$	-	$1$
$x$	+	$3$	-	$x^{2}$	-	$4 x$	-	$7$
			+	$x^{2}$	+	$3 x$
					-	$x$	-	$7$
					-	$x$	-	$3$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $- x - 3$

			-	$x$	-	$1$
$x$	+	$3$	-	$x^{2}$	-	$4 x$	-	$7$
			+	$x^{2}$	+	$3 x$
					-	$x$	-	$7$
					+	$x$	+	$3$

Step 5.12

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

			-	$x$	-	$1$
$x$	+	$3$	-	$x^{2}$	-	$4 x$	-	$7$
			+	$x^{2}$	+	$3 x$
					-	$x$	-	$7$
					+	$x$	+	$3$
							-	$4$

Step 5.13

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

$- x - 1 - \frac{4}{x + 3}$

Step 5.14

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

$y = - x - 1$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 3$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - x - 1$

Step 7