Precalculus Examples

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

Step 1

Find where the expression $\frac{(x + 2) {(x + 1)}^{3}}{(x + 2) {(x - 3)}^{2}}$ is undefined.

$x = - 2, x = 3$

Step 2

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

$x = 3$

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 $m$ .

$n = 4$

$m = 3$

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 $0$ .

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

Step 6.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x^{3}$ .

$x$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

Step 6.3

Multiply the new quotient term by the divisor.

$x$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

$x^{4}$

$4 x^{3}$

$3 x^{2}$

$18 x$

Step 6.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 4 x^{3} - 3 x^{2} + 18 x$

$x$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

$x^{4}$

$4 x^{3}$

$3 x^{2}$

$18 x$

Step 6.5

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

$x$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

$x^{4}$

$4 x^{3}$

$3 x^{2}$

$18 x$

$9 x^{3}$

$12 x^{2}$

$11 x$

Step 6.6

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

$x$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

$x^{4}$

$4 x^{3}$

$3 x^{2}$

$18 x$

$9 x^{3}$

$12 x^{2}$

$11 x$

$2$

Step 6.7

Divide the highest order term in the dividend $9 x^{3}$ by the highest order term in divisor $x^{3}$ .

$x$

$9$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

$x^{4}$

$4 x^{3}$

$3 x^{2}$

$18 x$

$9 x^{3}$

$12 x^{2}$

$11 x$

$2$

Step 6.8

Multiply the new quotient term by the divisor.

$x$

$9$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

$x^{4}$

$4 x^{3}$

$3 x^{2}$

$18 x$

$9 x^{3}$

$12 x^{2}$

$11 x$

$2$

$9 x^{3}$

$36 x^{2}$

$27 x$

$162$

Step 6.9

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{3} - 36 x^{2} - 27 x + 162$

$x$

$9$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

$x^{4}$

$4 x^{3}$

$3 x^{2}$

$18 x$

$9 x^{3}$

$12 x^{2}$

$11 x$

$2$

$9 x^{3}$

$36 x^{2}$

$27 x$

$162$

Step 6.10

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

$x$

$9$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

$x^{4}$

$5 x^{3}$

$9 x^{2}$

$7 x$

$2$

$x^{4}$

$4 x^{3}$

$3 x^{2}$

$18 x$

$9 x^{3}$

$12 x^{2}$

$11 x$

$2$

$9 x^{3}$

$36 x^{2}$

$27 x$

$162$

$48 x^{2}$

$16 x$

$160$

Step 6.11

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

$x + 9 + \frac{48 x^{2} + 16 x - 160}{x^{3} - 4 x^{2} - 3 x + 18}$

Step 6.12

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

$y = x + 9$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 3$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x + 9$

Step 8