Precalculus Examples

$\frac{2 x (x + 5)}{3 (x + 5)}$

Step 1

Find where the expression $\frac{2 x (x + 5)}{3 (x + 5)}$ is undefined.

$x = - 5$

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 $2 x$ out of $2 x^{2} + 10 x$ .

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Step 6.1.1.1

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

$\frac{2 x (x) + 10 x}{3 x + 15}$

Step 6.1.1.2

Factor $2 x$ out of $10 x$ .

$\frac{2 x (x) + 2 x (5)}{3 x + 15}$

Step 6.1.1.3

Factor $2 x$ out of $2 x (x) + 2 x (5)$ .

$\frac{2 x (x + 5)}{3 x + 15}$

Step 6.1.2

Factor $3$ out of $3 x + 15$ .

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Step 6.1.2.1

Factor $3$ out of $3 x$ .

$\frac{2 x (x + 5)}{3 (x) + 15}$

Step 6.1.2.2

Factor $3$ out of $15$ .

$\frac{2 x (x + 5)}{3 x + 3 \cdot 5}$

Step 6.1.2.3

Factor $3$ out of $3 x + 3 \cdot 5$ .

$\frac{2 x (x + 5)}{3 (x + 5)}$

Step 6.1.3

Cancel the common factor of $x + 5$ .

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Step 6.1.3.1

Cancel the common factor.

$\frac{2 x (x + 5)}{3 (x + 5)}$

Step 6.1.3.2

Rewrite the expression.

$\frac{2 x}{3}$

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 $0$ .

$3$

$2 x$

$0$

Step 6.3

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

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

Step 6.4

Multiply the new quotient term by the divisor.

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

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

$\frac{2 x}{3}$

Step 6.9

Since there is no polynomial portion from the polynomial division, there are no oblique asymptotes.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

No Oblique Asymptotes

Step 8