Question

How do you find the vertical, horizontal or slant asymptotes for `(5e^x)/((e^x)-6)`?

Answer 1

The vertical asymptote is `x=ln6`
The horizontal asymptote is `y=5` when `x in ]ln6, +oo[`
The horizontal asymptote is `y=0` when `x in ]-oo, ln6[`
No slant asymptote

Explanation:

Let `f(x)=(5e^x)/(e^x-6)`

The domain of `f(x)` is `D_f(x)=RR-{ln6}`

As you cannot divide by `0`, `e^x!=6`

The vertical asymptote is `x=ln6`

As the degree of the numerator is `=` to the degree of the denominator, there is no slant asymptote.

`lim_(x->+oo)f(x)=lim_(x->+oo)(5e^x)/e^x=5`

`lim_(x->-oo)f(x)=lim_(x->-oo)(5)/(1-6/e^x)=0^-`

The horizontal asymptote is `y=5` when `x in ]ln6, +oo[`

The horizontal asymptote is `y=0` when `x in ]-oo, ln6[`

graph{(5e^x)/(e^x-6) [-36.53, 36.54, -18.27, 18.28]}