Question

The value of \( \lim_{x \to 0} \frac{1 - \cos x}{x^{2}} \) is:

• A. 1
• B. 0
• C. 1/2
• D. 2

Hint:

For $0/0$ forms, L'Hopital's rule (differentiating numerator and denominator) is often faster.

Answer 1

The Correct Option is C

Step 1: Concept
Use the trigonometric identity $1 - \cos x = 2 \sin^{2}(x/2)$ or L'Hopital's Rule.
Step 2: Analysis
$\lim_{x \to 0} \frac{2 \sin^{2}(x/2)}{x^{2}} = \lim_{x \to 0} \frac{2 \sin^{2}(x/2)}{4(x/2)^{2}} = \frac{2}{4} \lim_{x \to 0} \left(\frac{\sin(x/2)}{x/2}\right)^{2}$.
Step 3: Conclusion
Since $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$, the result is $1/2 \times 1^{2} = 1/2$.
Final Answer: (C)