Precalculus Examples

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

Step 1

Find where the expression $\frac{(2 x + 3) (x - 4)}{3 (x - 4)}$ 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 $m$ .

$n = 2$

$m = 1$

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

Factor by grouping.

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Step 6.1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = - 5$ .

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Step 6.1.1.1.1

Factor $- 5$ out of $- 5 x$ .

$\frac{2 x^{2} - 5 (x) - 12}{3 x - 12}$

Step 6.1.1.1.2

Rewrite $- 5$ as $3$ plus $- 8$

$\frac{2 x^{2} + (3 - 8) x - 12}{3 x - 12}$

Step 6.1.1.1.3

Apply the distributive property.

$\frac{2 x^{2} + 3 x - 8 x - 12}{3 x - 12}$

Step 6.1.1.2

Factor out the greatest common factor from each group.

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Step 6.1.1.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + 3 x) - 8 x - 12}{3 x - 12}$

Step 6.1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (2 x + 3) - 4 (2 x + 3)}{3 x - 12}$

Step 6.1.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$\frac{(2 x + 3) (x - 4)}{3 x - 12}$

Step 6.1.2

Factor $3$ out of $3 x - 12$ .

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Step 6.1.2.1

Factor $3$ out of $3 x$ .

$\frac{(2 x + 3) (x - 4)}{3 (x) - 12}$

Step 6.1.2.2

Factor $3$ out of $- 12$ .

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

Step 6.1.2.3

Factor $3$ out of $3 x + 3 \cdot - 4$ .

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

Step 6.1.3

Cancel the common factor of $x - 4$ .

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Step 6.1.3.1

Cancel the common factor.

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

Step 6.1.3.2

Rewrite the expression.

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

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 $0$ .

$3$

$2 x$

$3$

Step 6.3

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $3$ .

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

Step 6.4

Multiply the new quotient term by the divisor.

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

Step 6.5

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

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

Step 6.6

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

		$\frac{2 x}{3}$
$3$		$2 x$	+	$3$
	-	$2 x$
		$0$

Step 6.7

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

		$\frac{2 x}{3}$
$3$		$2 x$	+	$3$
	-	$2 x$
		$0$	+	$3$

Step 6.8

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

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

Step 6.9

Split the solution into the polynomial portion and the remainder.

$\frac{2 x}{3} + 1$

Step 6.10

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

$y = \frac{2 x}{3}$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{2 x}{3}$

Step 8