Precalculus Examples

$f (x) = \frac{x^{5} - 3 x - 1}{x^{2} + 4}$

Step 1

Find where the expression $\frac{x^{5} - 3 x - 1}{x^{2} + 4}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

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 $m$ .

$n = 5$

$m = 2$

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 $0$ .

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

Step 6.2

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

$x^{3}$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

Step 6.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

$x^{5}$

$0$

$4 x^{3}$

Step 6.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{5} + 0 + 4 x^{3}$

$x^{3}$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

$x^{5}$

$0$

$4 x^{3}$

Step 6.5

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

$x^{3}$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

$x^{5}$

$0$

$4 x^{3}$

Step 6.6

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

$x^{3}$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

$x^{5}$

$0$

$4 x^{3}$

$0 x^{2}$

$3 x$

Step 6.7

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x^{2}$ .

$x^{3}$

$0 x^{2}$

$4 x$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

$x^{5}$

$0$

$4 x^{3}$

$0 x^{2}$

$3 x$

Step 6.8

Multiply the new quotient term by the divisor.

$x^{3}$

$0 x^{2}$

$4 x$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

$x^{5}$

$0$

$4 x^{3}$

$0 x^{2}$

$3 x$

$4 x^{3}$

$0$

$16 x$

Step 6.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{3} + 0 - 16 x$

$x^{3}$

$0 x^{2}$

$4 x$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

$x^{5}$

$0$

$4 x^{3}$

$0 x^{2}$

$3 x$

$4 x^{3}$

$0$

$16 x$

Step 6.10

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

$x^{3}$

$0 x^{2}$

$4 x$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

$x^{5}$

$0$

$4 x^{3}$

$0 x^{2}$

$3 x$

$4 x^{3}$

$0$

$16 x$

$13 x$

Step 6.11

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

$x^{3}$

$0 x^{2}$

$4 x$

$x^{2}$

$0 x$

$4$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$3 x$

$1$

$x^{5}$

$0$

$4 x^{3}$

$0 x^{2}$

$3 x$

$4 x^{3}$

$0$

$16 x$

$13 x$

$1$

Step 6.12

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

$x^{3} - 4 x + \frac{13 x - 1}{x^{2} + 4}$

Step 6.13

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

$y = x^{3} - 4 x$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = x^{3} - 4 x$

Step 8