How do you find the vertical, horizontal and slant asymptotes of: y = (x^2 + 7)/( 5x - 4x^2)?

Mar 9, 2017

The vertical asymptotes are $x = 0$ and $x = \frac{5}{4}$
The horizontal asymptote is $y = - \frac{1}{4}$
No slant asymptote

Explanation:

We cannot divide by $0$, so the denominator cannot $= 0$

So,

$5 x - 4 {x}^{2} \ne 0$

$x \left(5 - 4 x\right) \ne 0$

Therefore, $x \ne 0$ and $x \ne \frac{5}{4}$

The vertical asymptotes are $x = 0$ and $x = \frac{5}{4}$

The degree of the the numerator $=$ the degree of the denominator, there is no slant asymptote

${\lim}_{x \to \pm \infty} y = {\lim}_{x \to \pm \infty} - {x}^{2} / \left(4 {x}^{2}\right) = - \frac{1}{4}$

The horizontal asymptote is $y = - \frac{1}{4}$

graph{(y-(x^2+7)/(5x-4x^2))(y+1/4)(y-1000(x-5/4))=0 [-13.42, 14.3, -4.98, 8.89]}