Question

\(lim_{x\to 0}\) \(\frac{x({e}^{x}-1)}{1-cos x}\) =

• A. 0
• B.
• C. -2
• D. 2

Answer 1

The correct option is D

2

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