# How do you find the vertical, horizontal and slant asymptotes of: (x^2 - 5)/( x+3)?

Oct 26, 2016

#### Answer:

The vertical asymptote is $x = - 3$
The slant asymptote is $y = x - 3$
There is no horizontal asmptote

#### Explanation:

The domain of the function is $\mathbb{R} - \left(- 3\right)$
As we cannot divide by 0

As the degree of the numerator is greater than the degree of the denominator, we would expect a slant asymptote. So we make a long division

${x}^{2}$$\textcolor{w h i t e}{a a a a}$$- 5$∣$x + 3$
${x}^{2} + 3 x$$\textcolor{w h i t e}{a a}$∣$x - 3$
$0 - 3 x - 5$
$\textcolor{w h i t e}{a a}$$- 3 x - 9$
$\textcolor{w h i t e}{a a a a a}$$0 + 4$

So we can rewrite the function
$\frac{{x}^{2} - 5}{x + 3} = x - 3 + \frac{4}{x + 3}$

So the slant asymptote is $y = x - 3$