There are some rational functions that do a switcheroo when $x \to -\infty$ where (when graphed) they cross the horizontal asymptote in order to approach from the other side. It only seems to happen when the power is larger on bottom than the power on top.
I first came across this behavior in Algebra 2, and never found a satisfying answer to what was going on here. It's been nagging me since then as I went through Pre-Cal and AP Calculus, and never found any answers other than just a shrug and response of "that's just how it is."
Here's an example: $$y = \frac{x+1}{x^2}$$
When graphed, it looks like: 
Now that I've taken AP Calculus, I can also find that there's a relative minimum at $\left(-2, -\frac{1}{4}\right)$, but once again, it's unclear why. Can someone please shed some light on why these functions cross the horizontal asymptote?
The reason is because the horizontal asymptote is identically the $x$-axis for your function. Also, an asymptote only describes end behavior, so it's perfectly fine for the function to cross it somewhere near the origin. Since your function simultaneously has a root and has the $x$-axis as an asymptote, it will need to pull away from the axis, then slice through it, and then proceed eventually coming back to the axis.