Question

$\underset{x\to 0}{lim}\left[(1-e^x)\frac{\sin x}{\left|x\right|}\right],$ where $[.]$ represents greatest integer function, is equal to

• A. -1
• B. 1
• C. $lo{g}_{\sqrt{2}+1}(\sqrt{2}-1)$
• D. does not exist

Answer 1

The correct option is A -1

$\underset{x\to 0^{-}}{lim}\left[(1-e^x)\frac{\sin x}{\left|x\right|}\right]=\left[\right(1-e^{0^{-}}(-sinx/x)\left)\right]=[-k]=-1$ where $k\in (-1,0)$

$\underset{x\to 0^{+}}{lim}\left[(1-e^x)\frac{\sin x}{\left|x\right|}\right]=\left[\right(1-e^{0^{+}}\left)\right(sinx/x\left)\right)=[-l]=-1$ where $l\in (-1,0)$

Hence,

$\underset{x\to 0}{lim}\left[(1-e^x)\frac{\sin x}{\left|x\right|}\right]=-1$