1. A limit as $x \to \infty$ (or $x \to -\infty$) describes what happens when $x$ increases (or decreases) without bound.


2. The limit laws apply to limits as $x \to \pm \infty$, just as they apply to limits as $x \to a$ or $x \to a^+$ or $x \to a^-$.


3. An asymptote is a line that the graph $y=f(x)$ approaches. This line can be vertical, horizontal, or diagonal. A curve can be an asymptote too.


4. A horizontal asymptote is approached as $x \to \infty$ or as $x \to -\infty$. If $\displaystyle{\lim_{x \to \infty} f(x) = L}$ or $\displaystyle{\lim_{x \to -\infty} f(x) = L}$, then $y=L$ is a horizontal asymptote.


5. A vertical asymptote is approached as $y \to \infty$ or as $y \to -\infty$, which only happens if the function blows up at a finite value. If $\displaystyle{\lim_{x \to a^+} f(x) = \pm \infty}$ or $\displaystyle{\lim_{x \to a^-} f(x) = \pm \infty}$, then the function has a vertical asymptote at $x=a$.

You should recognize the following graphs and all of their asymptotes (if they have any).