Limit Continuity and Differentiability 1 Question 15
17. The value of $\lim _{x \rightarrow 0} \frac{\sqrt{\frac{1}{2}\left(1-\cos ^{2} x\right)}}{x}$ is
(1991, 2M)
(a) 1
(b) -1
(c) 0
(d) None of these
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Answer:
Correct Answer: 17. (d)
Solution:
- PLAN $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
Given, $\lim _{x \rightarrow 1} \frac{\sin (x-1)+a(1-x)}{(x-1)+\sin (x-1)} \frac{(1+\sqrt{x})(1-\sqrt{x})}{1-\sqrt{x}}=\frac{1}{4}$
$$ \begin{array}{ll} & \lim _{x \rightarrow 1} \frac{\frac{\sin (x-1)}{(x-1)}-a}{1+\frac{\sin (x-1)}{(x-1)}}=\frac{1}{4} \\ \Rightarrow \quad & \frac{1-a^{2}}{2}=\frac{1}{4} \Rightarrow(a-1)^{2}=1 \\ \Rightarrow \quad & a=2 \text { or } 0 \end{array} $$
Hence, the maximum value of $a$ is 2 .
