Precalculus Examples

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

Step 1

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

$x = 0$

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 $m$ .

$n = 4$

$m = 2$

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

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 6.1.1

Factor $x^{2}$ out of $x^{4}$ .

$\frac{x^{2} x^{2}}{2 x^{2}}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

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

$\frac{x^{2} x^{2}}{x^{2} \cdot 2}$

Step 6.1.2.2

Cancel the common factor.

$\frac{x^{2} x^{2}}{x^{2} \cdot 2}$

Step 6.1.2.3

Rewrite the expression.

$\frac{x^{2}}{2}$

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 $0$ .

$2$

$x^{2}$

$0 x$

$0$

Step 6.3

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

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

Step 6.4

Multiply the new quotient term by the divisor.

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

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

$\frac{x^{2}}{2}$

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