Precalculus Examples

$\frac{1 - x^{2}}{x - 2}$

Step 1

Find where the expression $\frac{1 - x^{2}}{x - 2}$ is undefined.

$x = 2$

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 $1$ as $1^{2}$ .

$\frac{1^{2} - x^{2}}{x - 2}$

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 $a = 1$ and $b = x$ .

$\frac{(1 + x) (1 - x)}{x - 2}$

Step 5.2

Expand $(1 + x) (1 - x)$ .

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Step 5.2.1

Apply the distributive property.

$\frac{1 (1 - x) + x (1 - x)}{x - 2}$

Step 5.2.2

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- x) + x (1 - x)}{x - 2}$

Step 5.2.3

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- x) + x \cdot 1 + x (- x)}{x - 2}$

Step 5.2.4

Remove parentheses.

$\frac{1 \cdot 1 + 1 \cdot (- 1 x) + x \cdot 1 + x (- x)}{x - 2}$

Step 5.2.5

Reorder $x$ and $1$ .

$\frac{1 \cdot 1 + 1 \cdot (- 1 x) + 1 \cdot x + x (- x)}{x - 2}$

Step 5.2.6

Remove parentheses.

$\frac{1 \cdot 1 + 1 \cdot (- 1 x) + 1 \cdot x + x \cdot (- 1 x)}{x - 2}$

Step 5.2.7

Reorder $x$ and $- 1$ .

$\frac{1 \cdot 1 + 1 \cdot (- 1 x) + 1 x - 1 \cdot (x \cdot x)}{x - 2}$

Step 5.2.8

Multiply $- 1$ by $x$ .

$\frac{1 \cdot 1 + 1 \cdot (- 1 x) + 1 x - 1 x \cdot x}{x - 2}$

Step 5.2.9

Multiply $1$ by $1$ .

$\frac{1 + 1 \cdot (- 1 x) + 1 x - 1 x \cdot x}{x - 2}$

Step 5.2.10

Multiply $- 1$ by $1$ .

$\frac{1 - x + 1 x - 1 x \cdot x}{x - 2}$

Step 5.2.11

Multiply $x$ by $1$ .

$\frac{1 - x + x - 1 x \cdot x}{x - 2}$

Step 5.2.12

Factor out negative.

$\frac{1 - x + x - (x \cdot x)}{x - 2}$

Step 5.2.13

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

$\frac{1 - x + x - (x \cdot x)}{x - 2}$

Step 5.2.14

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

$\frac{1 - x + x - (x \cdot x)}{x - 2}$

Step 5.2.15

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

$\frac{1 - x + x - x^{1 + 1}}{x - 2}$

Step 5.2.16

Add $1$ and $1$ .

$\frac{1 - x + x - x^{2}}{x - 2}$

Step 5.2.17

Add $- x$ and $x$ .

$\frac{1 + 0 - x^{2}}{x - 2}$

Step 5.2.18

Subtract $x^{2}$ from $0$ .

$\frac{1 - x^{2}}{x - 2}$

Step 5.2.19

Reorder $1$ and $- x^{2}$ .

$\frac{- x^{2} + 1}{x - 2}$

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 $0$ .

$x$

$2$

$x^{2}$

$0 x$

$1$

Step 5.4

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

				-	$x$
	$x$	-	$2$	-	$x^{2}$	+	$0 x$	+	$1$

Step 5.5

Multiply the new quotient term by the divisor.

			-	$x$
$x$	-	$2$	-	$x^{2}$	+	$0 x$	+	$1$
			-	$x^{2}$	+	$2 x$

Step 5.6

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

			-	$x$
$x$	-	$2$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$2 x$

Step 5.7

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

			-	$x$
$x$	-	$2$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$2 x$
					-	$2 x$

Step 5.8

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

			-	$x$
$x$	-	$2$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$2 x$
					-	$2 x$	+	$1$

Step 5.9

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

			-	$x$	-	$2$
$x$	-	$2$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$2 x$
					-	$2 x$	+	$1$

Step 5.10

Multiply the new quotient term by the divisor.

			-	$x$	-	$2$
$x$	-	$2$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$2 x$
					-	$2 x$	+	$1$
					-	$2 x$	+	$4$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x + 4$

			-	$x$	-	$2$
$x$	-	$2$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$2 x$
					-	$2 x$	+	$1$
					+	$2 x$	-	$4$

Step 5.12

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

			-	$x$	-	$2$
$x$	-	$2$	-	$x^{2}$	+	$0 x$	+	$1$
			+	$x^{2}$	-	$2 x$
					-	$2 x$	+	$1$
					+	$2 x$	-	$4$
							-	$3$

Step 5.13

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

$- x - 2 - \frac{3}{x - 2}$

Step 5.14

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

$y = - x - 2$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 2$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - x - 2$

Step 7