# How do you find the vertical, horizontal or slant asymptotes for (x² - 3x - 7)/(x+3) ?

Nov 3, 2016

The vertical asymptote is $x = 3$
and the slant asymptote is $y = x - 6$

#### Explanation:

As we cannot divide by zero, so $x \ne - 3$
$\therefore x = - 3$ is a vertical asymptote

As the degree of the numerator is greater than the degree of the denominator, we would expect a slant asymptote. W e have to do a long division.
$\textcolor{w h i t e}{a a a a}$${x}^{2} - 3 x - 7$$\textcolor{w h i t e}{a a a a}$∣$x + 3$
$\textcolor{w h i t e}{a a a a}$${x}^{2} + 3 x$$\textcolor{w h i t e}{a a a a a a a a}$∣$x - 6$
$\textcolor{w h i t e}{a a a a a}$$0 - 6 x - 7$
$\textcolor{w h i t e}{a a a a a a a}$$- 6 x - 18$
$\textcolor{w h i t e}{a a a a a a a a a}$$0 + 11$

Finally we have, $\frac{{x}^{2} - 3 x - 7}{x - 3} = x - 6 + \frac{11}{x + 3}$
The slant asymptote is $y = x - 6$

graph{(y-((x^2-3x-7)/(x+3)))(y-x+6)=0 [-58.5, 58.5, -29.27, 29.28]}