Limits Continuity And Differentiability Question 103
Question: What is $ \underset{x\to 0}{\mathop{\lim }}\frac{2(1-\cos x)}{x^{2}} $ equal to?
Options:
A) 0
B) 1/2
C) ¼
D) 1
Show Answer
Answer:
Correct Answer: D
Solution:
$ \underset{x\to 0}{\mathop{\lim }}\frac{2(1-\cos x)}{x^{2}}=\underset{x\to 0}{\mathop{\lim }}\frac{2.2{{\sin }^{2}}\frac{x}{2}}{x^{2}} $
$ =4\underset{x\to 0}{\mathop{\lim }}\frac{\sin \frac{x}{2}}{\frac{x}{2}\times 2}.\underset{x\to 0}{\mathop{\lim }}\frac{\sin \frac{x}{2}}{\frac{x}{2}\times 2} $
$ =\underset{x\to 0}{\mathop{\lim }}\frac{\sin \frac{x}{2}}{\frac{x}{2}}.\underset{x\to 0}{\mathop{\lim }}\frac{\sin \frac{x}{2}}{\frac{x}{2}}=1\times 1=1 $
BETA
