Question

For what values of x, if any, does `f(x) = x/(e^x-e^(2x))` have vertical asymptotes?

Answer 1

The function `x/(e^x-e^(2x))` does not have vertical asymptotes.

Explanation:

If we write the function as:

`f(x) = x/(e^x-e^(2x)) = x/(e^x(1-e^x))`

we can easily see that `f(x)` is continuous in all of `RR` except for the point `x=0` where the denominator vanishes.

Analysing the limit:

`lim_(x->0) f(x) = lim_(x->0)x/(e^x(1-e^x)) = lim_(x->0) 1/e^x * -1/(lim_(x->0) (e^x-1)/x) = 1*-1/1= -1`

Since

`lim (e^x-1)/x = 1`

So the function does not have vertical asymptotes.

graph{x/(e^x-e^(2x)) [-10, 10, -5, 5]}