Nilai \( \displaystyle \lim_{x\to 0} \ \frac{\tan x - \sin x}{x^3} = \cdots \)
Pembahasan:
\begin{aligned} \lim_{x\to 0} \ \frac{\tan x - \sin x}{x^3} &= \lim_{x\to 0} \ \frac{\frac{\sin x}{\cos x} - \sin x}{x^3} \\[8pt] &= \lim_{x\to 0} \ \frac{\sin x (1 - \cos x)}{x^3 \ \cos x} \\[8pt] &= \lim_{x\to 0} \ \frac{\sin x \ (2 \sin^2 \frac{1}{2}x)}{x^3} \\[8pt] &= 2 \cdot \lim_{x\to 0} \ \left( \frac{\sin x}{x} \cdot \frac{\sin \frac{1}{2}x}{x} \cdot \frac{\sin \frac{1}{2}x}{x} \cdot \frac{1}{\cos x} \right) \\[8pt] &= 2 \cdot 1 \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{\cos 0} = \frac{1}{2} \cdot \frac{1}{1} \\[8pt] &= \frac{1}{2} \end{aligned}