SATHEE: Limit Continuity and Differentiability 1 Question 15

Limit Continuity and Differentiability 1 Question 15

17. The value of $lim_{x \to 0} \frac{\sqrt{\frac{1}{2} (1 - \cos^{2} x)}}{x}$ is

(1991, 2M)

(a) 1

(b) -1

(d) None of these

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Answer:

Correct Answer: 17. (d)

Solution:

PLAN $lim_{x \to 0} \frac{\sin x}{x} = 1$

Given, $lim_{x \to 1} \frac{\sin (x - 1) + a (1 - x)}{(x - 1) + \sin (x - 1)} \frac{(1 + \sqrt{x}) (1 - \sqrt{x})}{1 - \sqrt{x}} = \frac{1}{4}$

$\begin{array}{ll} lim_{x \to 1} \frac{\frac{\sin (x - 1)}{(x - 1)} - a}{1 + \frac{\sin (x - 1)}{(x - 1)}} = \frac{1}{4} \\ \Rightarrow & \frac{1 - a^{2}}{2} = \frac{1}{4} \Rightarrow (a - 1)^{2} = 1 \\ \Rightarrow & a = 2 or 0 \end{array}$

Hence, the maximum value of $a$ is 2 .

Limit Continuity and Differentiability 1 Question 15

17. The value of $lim_{x \to 0} \frac{\sqrt{\frac{1}{2} (1 - \cos^{2} x)}}{x}$ is

Answer:

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Limit Continuity and Differentiability 1 Question 15

17. The value of limx→012(1−cos2⁡x)x is

Answer:

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17. The value of $lim_{x \to 0} \frac{\sqrt{\frac{1}{2} (1 - \cos^{2} x)}}{x}$ is