Precalculus Examples

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

Step 1

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

$x = 0$

Step 2

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 3

Find $n$ and $m$ .

$n = 4$

$m = 3$

Step 4

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

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

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Step 5.1

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

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 5.2

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

$2 x$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 5.3

Multiply the new quotient term by the divisor.

$2 x$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$2 x^{4}$

$0$

Step 5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} + 0 + 0 + 0$

$2 x$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$2 x^{4}$

$0$

Step 5.5

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

$2 x$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$2 x^{4}$

$0$

Step 5.6

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

$2 x$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$2 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$2 x^{4}$

$0$

$1$

Step 5.7

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

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

Step 5.8

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

$y = 2 x$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 2 x$

Step 7