Question

Evaluate $\underset{x\to 0}{lim}\left(\frac{(1+x{)}^{\frac{1}{x}}-e+\frac{ex}{2}}{{\sin }^{2}x}\right)$

Answer 1

Given expressions $\underset{x\to 0}{lim}\left(\frac{(1+x{)}^{1/x}-e+\frac{ex}{2}}{{\sin }^{2}x}\right)$

$\to \underset{x\to 0}{lim}\frac{(1+x\frac{1}{2}+\frac{1(1-x)}{2!}+\frac{1(1-x)(1-2x)}{2!}+....)-e+\frac{ex}{2}}{(\frac{\sin x}{x}{)}^2.x^2}$

$\to \underset{x\to 0}{lim}\left(\frac{e-e+\frac{ex}{2}}{x^2}\right)$

$\to$ Apply L-Hospital Rule

$\to \underset{x\to 0}{lim}\left(\frac{e/2}{2x}\right)$

$\to \underset{x\to 0}{lim}\left(0\right)=0$