Calculus I · Limits and Continuity · review

Squeeze and Trigonometric Limits Review

Section 3 Summary

Summary

• If $g\le f\le h$ and both outer limits equal $L$ , then the middle limit is $L$ . • Bounded oscillation multiplied by a shrinking factor is often squeezed to zero. • In radians, $\lim_{x\to0}\sin x/x=1$ and $\lim_{x\to0}\tan x/x=1$ . • Match the denominator to the trigonometric argument. • $\lim_{x\to0}(1-\cos x)/x^2=1/2$ .

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.

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