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

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Answer 1 · 2017-01-09T13:10:19

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

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#

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

So the function does not have vertical asymptotes.

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