Factoring, Rationalizing and Squeezing Limits — Cheat sheet

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Core rule

Use equality on nearby non-target points for factoring or rationalization. If $g\le f\le h$ nearby and $g,h\to L$ , then $f\to L$ .

\lim_{x\to0}\frac{\sin x}{x}=1,

\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12.

Angles must be in radians. Bounds must hold throughout a neighborhood. $0/0$ is a prompt for analysis, not an answer.