I'm assuming this function is: (2 + e^{1/x})/(2 - e^{1/x})?

Horizontal asymptotes occur when we take the function limit at x -> (+/-) infinity.

So, as x -> - infinity, 1/x -> 0 so e^{1/x} -> 1 Thus your function approaches (2 + 1)/(2 - 1) = 2. Since this is a constant, we have a horizontal asymptote at y = 2 for x->-infinity.

As x -> infinity we have the same behavior for 1/x so the function again has a horizontal asymptote at y = 2 for x->infinity.

(Technically there are stilltwohorizontal asymptotes, even though they are the same line.)

-Dan