If f (x) is the integral function of the function 2sin⁡x−sin⁡2xx3,x≠0, then limx→0 f′(x) is equal to

A
0
B
1
C
-1
D
2

Solution:

Since $f (x)$ is the integral function of $\frac{2 \sin x - \sin 2 x}{x^{3}}$

$∴ f^{'} (x) = \frac{2 \sin x - \sin 2 x}{x^{3}}$

Now,

$\begin{array}{l} lim_{x \to 0} f^{'} (x) = lim_{x \to 0} \frac{2 \sin x - \sin 2 x}{x^{3}} \\ \Rightarrow lim_{x \to 0} f^{'} (x) = lim_{x \to 0} \frac{2 \sin x}{x} \times \frac{1 - \cos x}{x^{2}} \\ \Rightarrow lim_{x \to 0} f^{'} (x) = 2 lim_{x \to 0} \frac{\sin x}{x} \times lim_{x \to 0} \frac{1 - \cos x}{x^{2}} = 2 \times \frac{1}{2} = 1 . \end{array}$

If $f (x)$ is the integral function of the function $\frac{2 \sin x - \sin 2 x}{x^{3}}, x \neq 0$ , then $lim_{x \to 0} f^{'} (x)$ is equal to

Solution:

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