Limits at Infinity and Horizontal Asymptotes

For the function $$h(x)=e^{x}$$, the function grows without bound, so no horizontal asymptote exists.

For the function $$g(x)=x*e^{-x}$$, although the limit is 0, the role of the polynomial factor complicates the demonstration of a horizontal asymptote.

For the function $$f(x)=e^{-x}$$, as $$x\to\infty$$ the value of the function approaches 0, revealing a horizontal asymptote at $$y=0$$.

For the function $$p(x)=\frac{e^x}{x}$$, the exponential numerator causes the function to diverge to infinity, eliminating a horizontal asymptote.