Question

$\underset{x\to 0}{lim}$ $\frac{x(e^x-1)}{1-cosx}$ =

• A. 0
• B. ∞
• C. -2
• D. 2

Answer 1

The correct option is D

2

$\underset{x\to 0}{lim}$ $\frac{x(e^x-1)}{1-cosx}$
=$\underset{x\to 0}{lim}$ $\frac{2x(e^x-1)}{4\left({sin}^2\frac{x}{2}\right)}$
=2 $\underset{x\to 0}{lim}$ [$\frac{{\left(\frac{x}{2}\right)}^2}{\left({sin}^2\frac{x}{2}\right)}$]($\frac{(e^x-1)}{x}$)
= 2