Precalculus Examples

$y = \frac{(x + 3) (x - 4) (x + 7)}{(x - 4)}$

Step 1

Find where the expression $\frac{(x + 3) (x - 4) (x + 7)}{x - 4}$ 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 $m$ .

$n = 3$

$m = 1$

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

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

$x$

$4$

$x^{3}$

$6 x^{2}$

$19 x$

$84$

Step 6.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$

Step 6.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			+	$x^{3}$	-	$4 x^{2}$

Step 6.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 4 x^{2}$

				$x^{2}$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$

Step 6.5

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$

Step 6.6

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$

Step 6.7

Divide the highest order term in the dividend $10 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$10 x$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$

Step 6.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$10 x$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$
					+	$10 x^{2}$	-	$40 x$

Step 6.9

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{2} - 40 x$

				$x^{2}$	+	$10 x$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$
					-	$10 x^{2}$	+	$40 x$

Step 6.10

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

				$x^{2}$	+	$10 x$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$
					-	$10 x^{2}$	+	$40 x$
							+	$21 x$

Step 6.11

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

				$x^{2}$	+	$10 x$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$
					-	$10 x^{2}$	+	$40 x$
							+	$21 x$	-	$84$

Step 6.12

Divide the highest order term in the dividend $21 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$10 x$	+	$21$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$
					-	$10 x^{2}$	+	$40 x$
							+	$21 x$	-	$84$

Step 6.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$10 x$	+	$21$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$
					-	$10 x^{2}$	+	$40 x$
							+	$21 x$	-	$84$
							+	$21 x$	-	$84$

Step 6.14

The expression needs to be subtracted from the dividend, so change all the signs in $21 x - 84$

				$x^{2}$	+	$10 x$	+	$21$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$
					-	$10 x^{2}$	+	$40 x$
							+	$21 x$	-	$84$
							-	$21 x$	+	$84$

Step 6.15

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

				$x^{2}$	+	$10 x$	+	$21$
$x$	-	$4$		$x^{3}$	+	$6 x^{2}$	-	$19 x$	-	$84$
			-	$x^{3}$	+	$4 x^{2}$
					+	$10 x^{2}$	-	$19 x$
					-	$10 x^{2}$	+	$40 x$
							+	$21 x$	-	$84$
							-	$21 x$	+	$84$
										$0$

Step 6.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 10 x + 21$

Step 6.17

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

$y = x^{2} + 10 x$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = x^{2} + 10 x$

Step 8