Question

$\underset{x\to 0}{lim}\frac{x(e^x-1)}{1-\cos x}$ is equal to

• A. $0$
• B. $\infty$
• C. $-2$
• D. $2$

Answer 1

The correct option is D $2$

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