When sketching this function, I determined that there's a horizontal asymptote at 0 but only by intuition and not by any mathematical means. I know that to find horizontal asymptotes, you must find $\displaystyle\lim_{x \to \infty}f(x)$ and $\displaystyle\lim_{x \to -\infty}f(x)$. What I'm guessing is that with $\displaystyle\lim_{x \to \infty}f(x)$, you basically get $(\infty)(\infty)$, meaning it shoots off up to infinity, but with $\displaystyle\lim_{x \to -\infty}f(x)$, it would be something like $\frac{\infty}{\infty}$, with the bottom growing faster than the top. Does this intuitively mean that as $x \to \infty$, $f(x) \to 0$? If not, how is this determined? I feel like I should know this but for some reason the answer isn't coming to mind.

Section 5.5 # 5