Question

The value of $\underset{x\to 0}{lim}\{\left[\frac{100x}{\sin x}\right]+\left[\frac{99\sin x}{x}\right]\}$,where [.] represents the greatest integer function, is:

• A. $199$
• B. $198$
• C. $0$
• D. $noneofthese$

Answer 1

Answer: (B)

$198$

From the graph of x and sin x, we know that for all x > 0,

$x>sinx$ or $\frac{x}{sinx}>1$

Therefore, $\frac{sinx}{x}<1$

Thus, $100<\frac{100x}{sinx}<101$

And , $98<\frac{99sinx}{x}<99$

Thus, $\left[\frac{100x}{sinx}\right]=100$

And, $\left[\frac{99sinx}{x}\right]=98$

Hence, their sum is $100+98=198$