_ x 0 , ( x - ,x x ) , , ( 1 x ) - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to 0} \,\left( {\frac{{x - \sin \,x}}{x}} \right)\,\sin \,\left( {\frac{1}{x}} \right)$

A
equals $1$
✓
equals $0$
C
does not exist
D
equals $- 1$

✓

Answer

Correct option: B.

equals $0$

b
Consider $\,\,\mathop {\lim }\limits_{x \to 0} \left( {\frac{{x - \sin x}}{x}} \right)\sin \left( {\frac{1}{x}} \right)$

$\, = \,\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{x\left( {1 - \frac{{\sin x}}{x}} \right)}}{x}} \right] \times \mathop {\lim }\limits_{x \to 0} \sin \left( {\frac{1}{x}} \right)$

$ = \mathop {\lim }\limits_{x \to 0} \left[ {1 - \frac{{\sin x}}{x}} \right] \times \mathop {\lim }\limits_{x \to 0} \sin \left( {\frac{1}{x}} \right)$

$ = \left[ {1 - \mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x}} \right] \times \mathop {\lim }\limits_{x \to 0} \sin \left( {\frac{1}{x}} \right)$

$ = 0 \times \mathop {\lim }\limits_{x \to 0} \sin \left( {\frac{1}{x}} \right) = 0$

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