Mr. Integral
Math notes/Calculus I/Limits & Continuity

Limits and One-Sided Behavior — Cheat sheet

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Core rule

A finite two-sided limit exists exactly when its two one-sided limits agree. The value at the point need not equal the limit.

lim⁡x→af(x)=L  ⟺  lim⁡x→a−f(x)=lim⁡x→a+f(x)=L.\lim_{x\to a}f(x)=L\iff\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L.

Watch for

0/00/0 is an indeterminate form. Simplify on a punctured neighborhood; do not substitute a value into a cancelled original denominator. Tables and plots suggest limits but do not prove them.