Find the Asymptotes f(x)=(x^2+6x-9)/(x-2)

$f (x) = \frac{x^{2} + 6 x - 9}{x - 2}$

Find where the expression $\frac{x^{2} + 6 x - 9}{x - 2}$ is undefined.

$x = 2$

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).

Find $n$ and $m$ .

$n = 2$

$m = 1$

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

No Horizontal Asymptotes

Find the oblique asymptote using polynomial division.

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

$x$

$2$

$x^{2}$

$6 x$

$9$

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

					$x$
	$x$	-	$2$		$x^{2}$	+	$6 x$	-	$9$

Multiply the new quotient term by the divisor.

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

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

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

Divide the highest order term in the dividend $8 x$ by the highest order term in divisor $x$ .

Multiply the new quotient term by the divisor.

The expression needs to be subtracted from the dividend, so change all the signs in $8 x - 16$

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

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

$x + 8 + \frac{7}{x - 2}$

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

$y = x + 8$

This is the set of all asymptotes.

Vertical Asymptotes: $x = 2$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x + 8$