Calculus I · Limits and Continuity · reference

Trigonometric Limit Decision Guide

Trig-Limit Decision Guide

Method

For a trigonometric limit near zero:

• Substitute first. • If the result is $0/0$ , look for $\sin u/u$ , $\tan u/u$ , or $(1-\cos u)/u^2$ . • Multiply by constants to make the denominator match the trigonometric argument. • Use identities such as $1-\cos^2u=\sin^2u$ when needed. • If a bounded trig factor is multiplied by something tending to zero, consider the Squeeze Theorem.

After the explanation

Use the section idea

Reading lens

Can the expression be rewritten around a known small-angle limit, with every scaling factor accounted for?

Mental model

The fundamental sine limit is a reusable local shape: other trigonometric limits work when you expose that shape through identities and scaling.

Decision

Look for a bounded oscillation times a shrinking factor, or rewrite the expression into sine-over-angle factors whose arguments match their denominators.

Common trap

The sine function is not equal to its angle; their ratio merely approaches one near zero, and that statement requires radian measure.

Check yourself

Mastery means you can mark every scaling factor before simplifying and can explain where the Squeeze Theorem enters the argument.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.