Precalculus Examples

$f (x) = \frac{10 x^{3} - 15 x^{2} + x - 1}{5 x^{2} - 2}$

Step 1

Find where the expression $\frac{10 x^{3} - 15 x^{2} + x - 1}{5 x^{2} - 2}$ is undefined.

$x = - \frac{\sqrt{10}}{5}, x = \frac{\sqrt{10}}{5}$

Step 2

Since $\frac{10 x^{3} - 15 x^{2} + x - 1}{5 x^{2} - 2}$ $\to$ $- \infty$ as $x$ $\to$ $- \frac{\sqrt{10}}{5}$ from the left and $\frac{10 x^{3} - 15 x^{2} + x - 1}{5 x^{2} - 2}$ $\to$ $\infty$ as $x$ $\to$ $- \frac{\sqrt{10}}{5}$ from the right, then $x = - \frac{\sqrt{10}}{5}$ is a vertical asymptote.

$x = - \frac{\sqrt{10}}{5}$

Step 3

Since $\frac{10 x^{3} - 15 x^{2} + x - 1}{5 x^{2} - 2}$ $\to$ $\infty$ as $x$ $\to$ $\frac{\sqrt{10}}{5}$ from the left and $\frac{10 x^{3} - 15 x^{2} + x - 1}{5 x^{2} - 2}$ $\to$ $- \infty$ as $x$ $\to$ $\frac{\sqrt{10}}{5}$ from the right, then $x = \frac{\sqrt{10}}{5}$ is a vertical asymptote.

$x = \frac{\sqrt{10}}{5}$

Step 4

List all of the vertical asymptotes:

$x = - \frac{\sqrt{10}}{5}, \frac{\sqrt{10}}{5}$

Step 5

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 6

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 7

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

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Step 8.1

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

$5 x^{2}$

$0 x$

$2$

$10 x^{3}$

$15 x^{2}$

$x$

$1$

Step 8.2

Divide the highest order term in the dividend $10 x^{3}$ by the highest order term in divisor $5 x^{2}$ .

$2 x$

$5 x^{2}$

$0 x$

$2$

$10 x^{3}$

$15 x^{2}$

$x$

$1$

Step 8.3

Multiply the new quotient term by the divisor.

$2 x$

$5 x^{2}$

$0 x$

$2$

$10 x^{3}$

$15 x^{2}$

$x$

$1$

$10 x^{3}$

$0$

$4 x$

Step 8.4

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{3} + 0 - 4 x$

						$2 x$
$5 x^{2}$	+	$0 x$	-	$2$		$10 x^{3}$	-	$15 x^{2}$	+	$x$	-	$1$
					-	$10 x^{3}$	-	$0$	+	$4 x$

Step 8.5

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

$2 x$

$5 x^{2}$

$0 x$

$2$

$10 x^{3}$

$15 x^{2}$

$x$

$1$

$10 x^{3}$

$0$

$4 x$

$15 x^{2}$

$5 x$

Step 8.6

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

						$2 x$
$5 x^{2}$	+	$0 x$	-	$2$		$10 x^{3}$	-	$15 x^{2}$	+	$x$	-	$1$
					-	$10 x^{3}$	-	$0$	+	$4 x$
							-	$15 x^{2}$	+	$5 x$	-	$1$

Step 8.7

Divide the highest order term in the dividend $- 15 x^{2}$ by the highest order term in divisor $5 x^{2}$ .

						$2 x$	-	$3$
$5 x^{2}$	+	$0 x$	-	$2$		$10 x^{3}$	-	$15 x^{2}$	+	$x$	-	$1$
					-	$10 x^{3}$	-	$0$	+	$4 x$
							-	$15 x^{2}$	+	$5 x$	-	$1$

Step 8.8

Multiply the new quotient term by the divisor.

						$2 x$	-	$3$
$5 x^{2}$	+	$0 x$	-	$2$		$10 x^{3}$	-	$15 x^{2}$	+	$x$	-	$1$
					-	$10 x^{3}$	-	$0$	+	$4 x$
							-	$15 x^{2}$	+	$5 x$	-	$1$
							-	$15 x^{2}$	+	$0$	+	$6$

Step 8.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{2} + 0 + 6$

						$2 x$	-	$3$
$5 x^{2}$	+	$0 x$	-	$2$		$10 x^{3}$	-	$15 x^{2}$	+	$x$	-	$1$
					-	$10 x^{3}$	-	$0$	+	$4 x$
							-	$15 x^{2}$	+	$5 x$	-	$1$
							+	$15 x^{2}$	-	$0$	-	$6$

Step 8.10

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

						$2 x$	-	$3$
$5 x^{2}$	+	$0 x$	-	$2$		$10 x^{3}$	-	$15 x^{2}$	+	$x$	-	$1$
					-	$10 x^{3}$	-	$0$	+	$4 x$
							-	$15 x^{2}$	+	$5 x$	-	$1$
							+	$15 x^{2}$	-	$0$	-	$6$
									+	$5 x$	-	$7$

Step 8.11

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

$2 x - 3 + \frac{5 x - 7}{5 x^{2} - 2}$

Step 8.12

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

$y = 2 x - 3$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{\sqrt{10}}{5}, \frac{\sqrt{10}}{5}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 2 x - 3$

Step 10