Asymptotes, Squeeze Theorem and the IVT

Limits at infinity

For a rational function compare degrees: bottom-heavy $\Rightarrow$ limit $0$; equal degrees $\Rightarrow$ ratio of leading coefficients; top-heavy $\Rightarrow$ $\pm\infty$ (no horizontal asymptote).

With a square root, factor the highest power out of the radical and remember $\sqrt{x^2}=|x|$, which equals $-x$ as $x\to-\infty$. That sign is the single most-tested detail of the topic.

Squeeze Theorem

If $g(x)\le f(x)\le h(x)$ near $c$ and $\lim_{x\to c}g=\lim_{x\to c}h=L$, then $\lim_{x\to c}f=L$. Its signature use: anything bounded times something going to zero, e.g. $x^2\sin\frac1x$.

Intermediate Value Theorem

If $f$ is continuous on the closed interval $[a,b]$ and $k$ lies between $f(a)$ and $f(b)$, then $f(c)=k$ for at least one $c\in(a,b)$. The hypothesis of continuity must be stated explicitly to earn the point.