Proof of Trig Limits — Question 8

Question 8

Prove that $$\lim_{x\to 0}\frac{\cos x - 1}{x}=0.$$

Question 8 - Solution

We begin by rewriting the expression using a trigonometric identity.

Recall that $$\cos x - 1 = -\bigl(1-\cos x\bigr).$$

Thus, $$\frac{\cos x - 1}{x} = -\frac{1-\cos x}{x}.$$

From a previously established trigonometric limit, $$\lim_{x\to 0}\frac{1-\cos x}{x}=0.$$

Therefore, $$\lim_{x\to 0}\frac{\cos x - 1}{x} = -\lim_{x\to 0}\frac{1-\cos x}{x} = -0 = 0.$$

$$\boxed{0}$$