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

Answer 1

Ezra Adams

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#

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]}

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Answer 2

Emma Aldridge

The function f(x) = x/(e^x-e^(2x)) has vertical asymptotes at x = 0 and x = ln(2).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.