Precalculus Examples

$f (x) = \frac{x^{2} - 25}{x}$

Step 1

Find where the expression $\frac{x^{2} - 25}{x}$ 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 $m$ .

$n = 2$

$m = 1$

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

Simplify the numerator.

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Step 5.1.1

Rewrite $25$ as $5^{2}$ .

$\frac{x^{2} - 5^{2}}{x}$

Step 5.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{(x + 5) (x - 5)}{x}$

Step 5.2

Expand $(x + 5) (x - 5)$ .

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Step 5.2.1

Apply the distributive property.

$\frac{x (x - 5) + 5 (x - 5)}{x}$

Step 5.2.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 5 + 5 (x - 5)}{x}$

Step 5.2.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 5 + 5 x + 5 \cdot - 5}{x}$

Step 5.2.4

Reorder $x$ and $- 5$ .

$\frac{x \cdot x - 5 x + 5 x + 5 \cdot - 5}{x}$

Step 5.2.5

Raise $x$ to the power of $1$ .

$\frac{x \cdot x - 5 x + 5 x + 5 \cdot - 5}{x}$

Step 5.2.6

Raise $x$ to the power of $1$ .

$\frac{x \cdot x - 5 x + 5 x + 5 \cdot - 5}{x}$

Step 5.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} - 5 x + 5 x + 5 \cdot - 5}{x}$

Step 5.2.8

Add $1$ and $1$ .

$\frac{x^{2} - 5 x + 5 x + 5 \cdot - 5}{x}$

Step 5.2.9

Multiply $5$ by $- 5$ .

$\frac{x^{2} - 5 x + 5 x - 25}{x}$

Step 5.2.10

Add $- 5 x$ and $5 x$ .

$\frac{x^{2} + 0 - 25}{x}$

Step 5.2.11

Subtract $25$ from $0$ .

$\frac{x^{2} - 25}{x}$

Step 5.3

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

$x$

$0$

$x^{2}$

$0 x$

$25$

Step 5.4

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

					$x$
	$x$	+	$0$		$x^{2}$	+	$0 x$	-	$25$

Step 5.5

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	-	$25$
			+	$x^{2}$	+	$0$

Step 5.6

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

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	-	$25$
			-	$x^{2}$	-	$0$

Step 5.7

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

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	-	$25$
			-	$x^{2}$	-	$0$
						$0$

Step 5.8

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

				$x$
$x$	+	$0$		$x^{2}$	+	$0 x$	-	$25$
			-	$x^{2}$	-	$0$
						$0$	-	$25$

Step 5.9

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

$x - \frac{25}{x}$

Step 5.10

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

$y = x$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x$

Step 7