Finding the horizontal asymptote of a composite function given the two functions that make it up

Question

Given a graph of two functions - f(x) and g(x) and told that h(x) = f(x)/g(x), how does one go about finding different limits of h, and how does one determine where f has a horizontal asymptote?
Also, in general, if a graph can cross a horizontal asymptote, then what does it do? Or, is it that the graph can cross it but will stay right next to it?

Answer 1

As for horizontal asymptotes, in general, one may seek, as by @Michael Rybkin, the limits: $$\lim_{x\to\pm\infty}h(x).$$ In case, e.g. $\lim\limits_{x\to\infty}f(x)=a$, then the horizontal line $y=a$ is the (unique) horizontal asymptote of $f$ at $+\infty$.

AS for intersecting ("crossing") a horizontal asymptote, this actually implies nothing. I suspect that you may have some kind of "monotonic" view of asymptotic behaviour, such as the following ($f(x)=1-e^{-x}$):

However, one may have cases like the following one, where there is a non-monotonic behaviour (here $f(x)=\frac{\sin(5x)}{x}$):