Nilai \( \displaystyle \lim_{x\to 0} \ \frac{\sec x - 1}{x} = \cdots \)
Pembahasan:
Ingat: \( \displaystyle \sec x = \frac{1}{\cos x} \) dan \( \displaystyle \lim_{x\to 0} \ \frac{1-\cos x}{x} = 0 \)
Dengan demikian, penyelesaian dari limit ini, yaitu:
\begin{aligned} \lim_{x\to 0} \ \frac{\sec x - 1}{x} &= \lim_{x\to 0} \ \frac{\frac{1}{\cos x} - 1}{x} \\[8pt] &= \lim_{x\to 0} \ \frac{1-\cos x}{x \ \cos x} \\[8pt] &= \lim_{x\to 0} \ \frac{1}{\cos x} \cdot \lim_{x\to 0} \ \frac{1-\cos x}{x} \\[8pt] &= \frac{1}{\cos 0} \cdot 0 = \frac{1}{1} \cdot 0 \\[8pt] &= 0 \end{aligned}