Question

$\underset{x\to 0}{lim}\left[m\frac{\sin x}{x}\right]$ is equal to (where $mϵI$ and $[.]$ denotes greatest integer function.)

• A. $m,ifm\le 0$
• B. $m-1,ifm>0$
• C. $m-1,ifm<0$
• D. $m,ifm>0$

Answer 1

Answer: (A), (B)

The correct options are
A $m,ifm\le 0$
B $m-1,ifm>0$

$[m\times \frac{sin\left(x\right)}{x}]=[m\times \frac{(x-\frac{x^3}{3!}+\frac{x^5}{5!}+.....)}{x}]=[m\times (1-\frac{x^2}{3!}+\frac{x^4}{5!}+...\left)\right]$

$=mifm<0orm-1ifm>0$
Greatest integer function examples $\left[5\right]=5,[-2.7]=-3,\left[2.7\right]=2$
First expand $sin\left(x\right)$ using Taylor series and use the concepts of greatest integer function.
Hence, options 'A' and 'B' are correct.