Calculus – How to Evaluate Limit of (tan(x) – sin(x)) / x(1 – cos(x)) as x Approaches 0

Question

How to evaluate $\lim_{x \to 0} \frac{\tan{x} – \sin{x}}{x(1 – \cos{x})}$ ?

The limit is of the form $0/0$, so I thought we can apply L'hopital. But the limit of the derivatives is also $0/0$, and the limit of the second derivatives is also $0/0$. Is there something else we need to do ?

Answer 1

$\frac{\tan{x} - \sin{x}}{x(1 - \cos{x})}=\frac {\sin x} x\frac {\frac 1 {\cos x}-1} {1-\cos x}=\frac {\sin x} x \frac 1 {\cos x}$ so the limit is $1$.