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Question: The value of \(\lim_{x \rightarrow 0}\left\{ \left\lbrack \frac{100x}{\sin x} \right\rbrack + \left\...

The value of lim⁡x→0{[100xsin⁡x]+[99sin⁡xx]}\lim_{x \rightarrow 0}\left\{ \left\lbrack \frac{100x}{\sin x} \right\rbrack + \left\lbrack \frac{99\sin x}{x} \right\rbrack \right\}

where [ ] represents the greatest function is-

A

199

B

198

C

0

D

197

Answer

198

Explanation

Solution

We know that

x > sin x for all x > 0

and x < sin x for all x < 0

Ž xsin⁡x\frac{x}{\sin x}> 1 and sin⁡xx\frac{\sin x}{x}< 1 for x ® 0

Ž 100xsin⁡x\frac{100x}{\sin x}> 100 and 99sin⁡xx\frac{99\sin x}{x}< 99

\ lim⁡x→0([100xsin⁡x]+[99sin⁡xx])\lim_{x \rightarrow 0}\left( \left\lbrack \frac{100x}{\sin x} \right\rbrack + \left\lbrack \frac{99\sin x}{x} \right\rbrack \right)

= lim⁡x→0\lim_{x \rightarrow 0} (100 +98) = 198