Calculus I · Limits and Continuity · lesson

Why sin(x)/x Approaches 1

Concept

Learning objectives

Understand geometrically why $\lim_{x\to0}\sin x/x=1$ when angles are measured in radians, and use it to evaluate scaled trigonometric limits.

The Fundamental Trigonometric Limit

The most important trigonometric limit in first-semester calculus is

\boxed{\lim_{x\to0}\frac{\sin x}{x}=1}.

It is not a random formula. It follows from the geometry of the unit circle and the Squeeze Theorem.

Common mistake

The formula requires radians. If $x$ is measured in degrees, $\sin x/x$ approaches $\pi/180$ , not $1$ . Calculus uses radians because arc length on the unit circle equals the angle measure itself.

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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