How do you find the asymptotes for #y = (4 e^x)/(e^x - 2)#?

Answer 1

Vertical asymptote at #x=ln2#.
Horizontal asymptote at #y=4#.

There exists vertical asymptotes at points that set the denominator to zero, that is, when #e^x-2=0# or when #e^x=2# #=>x=ln2#.
Horizontal asymptotes occur at #y=lim_(x->+-oo)[(4e^x)/(e^x-2)]=4#

The graph of the function verifies this.

graph{(4e^x)/(e^x-2) [-10.59, 11.91, -4.14, 7.11]}

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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.

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.

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.

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.

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