Question

How do you find the asymptotes for `y = (5e^x)/ ((e^x) - 8)`?

Answer 1

Vertical asymptote at `x = ln8`
Horizontal asymptotes at: `y = 0` and `y = 5`

Explanation:

The vertical asymptote is found when `D(x) = 0`:
`e^x - 8 = 0` so `e^x = 8`

Solve for `x` by logging both sides of your equation:

`ln e^x = ln 8`

Vertical asymptote at `x = ln8`

Finding horizontal asymptotes :

1. `N(x) = 5e^x = 0; e^x = 0; ln e^x = ln 0; x = ln 0` which is undefined. By definition logarithms are positive: `x > 0` This means there is a horizontal asymptote at `y = 0`

2. `lim x->oo (5e^x/e^x)/(e^x/e^x - 8/e^x) = 5/(1-0) = 5` so there is a horizontal asymptote at `y = 5`

Summary:
Vertical asymptote at `x = ln8`
Horizontal asymptotes at: `y = 0` and `y = 5`