Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `(6e^x)/(e^x-8)`?

Answer 1

The vertical asymptote is `x=ln8`
The horizontal asymptotes are `y=0` and `y=6`
No oblique asymptote

Explanation:

As you cannot divide by `0`

The denominator must be `!=0`

`e^x-8!=0`

`e^x!=8`

`x!=ln8`

So the vertical asymptote is `x=ln8`

Let `f(x)=(6e^x)/(e^x-8)`

`f(x)=(6e^x)/(e^x-8)=6/(1-8e^(-x))`

`lim_(x->-oo)f(x)=lim_(x->-oo)6/((1-8e^(-x)))=6/-oo=0^(-)`

`lim_(x->+oo)f(x)=lim_(x->+oo)6/((1-8e^(-x)))=6`

The horizontal asymptotes are `y=0` and `y=6`

graph{(6e^x)/(e^x-8) [-15.04, 16.99, -5.05, 10.96]}