What is the horizontal asymptote of ##y=e^x## ?

The horizontal asymptote of ##y=e^x## is ##y=0##.

The graph of ##y=e^x## is asymptotic to the negative x-axis only. Horizontal asymptotes of a graph are related to the end behavior of a function. Since

##lim_(x->-oo)e^x=0##

the horizontal asymptote is ##y=0##.

This is because, for negative values of x,

##e^x=1/(e^|x|##

And, as ##x->-oo##, ##1/e^|x|## will approach zero.

See image below for a graphical interpretation.

Notice that,

##lim_(x->oo)e^x=oo##

So, as increases without bound, so does y.