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.