AP Calculus AB and BC glossary

Special trigonometric limits

Also called: Fundamental trigonometric limits

The special trigonometric limits are sine of x over x approaching 1 and one minus cosine of x over x approaching 0, both as x approaches zero. They hold only when x is measured in radians, and they are what let you differentiate sine and cosine from the definition.

\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{x \to 0} \frac{1 - \cos x}{x} = 0

Direct substitution turns each of these into $\frac{0}{0}$ , an indeterminate form, so you cannot read them off by plugging in. The squeeze theorem proves the first, and the pair is what lets the limit definition deliver $f'(x) = \cos x$ for $f(x) = \sin x$ .

To evaluate a $\frac{0}{0}$ trig limit, force the argument of the sine to match the denominator. For example $\lim_{x \to 0}\frac{\sin 5x}{2x} = \lim_{x \to 0}\frac{5}{2}\cdot\frac{\sin 5x}{5x} = \frac{5}{2}$ , since $\frac{\sin 5x}{5x} \to 1$ .

The mistake

Two errors dominate. First, applying these in degrees: the clean values $1$ and $0$ hold only in radians, and in degrees $\lim_{x \to 0}\frac{\sin x}{x} = \frac{\pi}{180}$ . Second, forgetting that the argument and the denominator must match, so $\lim_{x \to 0}\frac{\sin 5x}{x} = 5$ , not $1$ .

Appears in: Unit 1: Limits and Continuity