# How do you find the slant asymptote of y=((4x^3)+(x^2)+x+4)/((x^2)+5x)?

Feb 20, 2016

Slant asymptote is $y = \left(4 x - 19\right)$

#### Explanation:

While vertical asymptotes are given by solution of $\left({x}^{2} + 5 x\right) = 0$ i.e. $x \left(x + 5\right) = 0$ i.e. they are $x = 0$ and $x = - 5$

To find slant asymptote of $y = \frac{4 {x}^{3} + {x}^{2} + x + 4}{{x}^{2} + 5 x}$, divide $\left(4 {x}^{3} + {x}^{2} + x + 4\right)$ by $\left({x}^{2} + 5 x\right)$, o

$4 x \left({x}^{2} + 5 x\right) - 19 \left({x}^{2} + 5 x\right) + 96 x + 4$ i.e.

$y = \left(4 x - 19\right) + \frac{96 x + 4}{{x}^{2} + 5 x}$

Hence slant asymptote is $y = \left(4 x - 19\right)$