Find the Asymptotes f(x)=(x^2-x-6)/(x-8)

$f (x) = \frac{x^{2} - x - 6}{x - 8}$

Step 1

Find where the expression $\frac{x^{2} - x - 6}{x - 8}$ is undefined.

$x = 8$

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.

Factor $x^{2} - x - 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

$\frac{(x - 3) (x + 2)}{x - 8}$

Expand $(x - 3) (x + 2)$ .

Apply the distributive property.

$\frac{x (x + 2) - 3 (x + 2)}{x - 8}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 - 3 (x + 2)}{x - 8}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 - 3 x - 3 \cdot 2}{x - 8}$

Reorder $x$ and $2$ .

$\frac{x \cdot x + 2 x - 3 x - 3 \cdot 2}{x - 8}$

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

$\frac{x \cdot x + 2 x - 3 x - 3 \cdot 2}{x - 8}$

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

$\frac{x \cdot x + 2 x - 3 x - 3 \cdot 2}{x - 8}$

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

$\frac{x^{1 + 1} + 2 x - 3 x - 3 \cdot 2}{x - 8}$

Add $1$ and $1$ .

$\frac{x^{2} + 2 x - 3 x - 3 \cdot 2}{x - 8}$

Multiply $- 3$ by $2$ .

$\frac{x^{2} + 2 x - 3 x - 6}{x - 8}$

Subtract $3 x$ from $2 x$ .

$\frac{x^{2} - x - 6}{x - 8}$

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

$x$

$8$

$x^{2}$

$x$

$6$

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

					$x$
	$x$	-	$8$		$x^{2}$	-	$x$	-	$6$

Multiply the new quotient term by the divisor.

				$x$
$x$	-	$8$		$x^{2}$	-	$x$	-	$6$
			+	$x^{2}$	-	$8 x$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} - 8 x$

				$x$
$x$	-	$8$		$x^{2}$	-	$x$	-	$6$
			-	$x^{2}$	+	$8 x$

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

				$x$
$x$	-	$8$		$x^{2}$	-	$x$	-	$6$
			-	$x^{2}$	+	$8 x$
					+	$7 x$

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

				$x$
$x$	-	$8$		$x^{2}$	-	$x$	-	$6$
			-	$x^{2}$	+	$8 x$
					+	$7 x$	-	$6$

Divide the highest order term in the dividend $7 x$ by the highest order term in divisor $x$ .

				$x$	+	$7$
$x$	-	$8$		$x^{2}$	-	$x$	-	$6$
			-	$x^{2}$	+	$8 x$
					+	$7 x$	-	$6$

Multiply the new quotient term by the divisor.

				$x$	+	$7$
$x$	-	$8$		$x^{2}$	-	$x$	-	$6$
			-	$x^{2}$	+	$8 x$
					+	$7 x$	-	$6$
					+	$7 x$	-	$56$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x - 56$

				$x$	+	$7$
$x$	-	$8$		$x^{2}$	-	$x$	-	$6$
			-	$x^{2}$	+	$8 x$
					+	$7 x$	-	$6$
					-	$7 x$	+	$56$

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

				$x$	+	$7$
$x$	-	$8$		$x^{2}$	-	$x$	-	$6$
			-	$x^{2}$	+	$8 x$
					+	$7 x$	-	$6$
					-	$7 x$	+	$56$
							+	$50$

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

$x + 7 + \frac{50}{x - 8}$

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

$y = x + 7$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 8$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x + 7$

Step 7

Find the Asymptotes f(x)=(x^2-x-6)/(x-8)

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