What are the equations of the horizontal asymptotes of y = (4e^x+7)/(e^x-1) in the xy plane?

In order to find a horizontal asymptote of a function f(x), you need to calculate
= n
where n is a finite number.

in your case you have to caculate separately the two limits:

a) = 

To solve this, you need to regroup eˣ both on the numerator and on the denominator:
= 

At this point the two eˣ cancel out and the parenthesis tend to 1, therefore
= 4
Your first horizontal asymptote is y = 4.

b) Let's calcuate now:

=  = -7
Therefore your second horizontal asymptote is y = -7

Your answer is, hence, y = 4 and y = -7 are horizontal asymptotes of the given function.