Question

The value of $\displaystyle\lim _{x \rightarrow 0} \frac{(1+x)^{1 / x}-e+\frac{1}{2} e x}{x^{2}}$ is

• A. $\frac{-11 e}{24}$
• B. $\frac{e}{11}$
• C. $\frac{e}{24}$
• D. $\frac{11 e}{24}$
• E. None of these

Answer 1

Answer: (D)

$\displaystyle\lim _{x \rightarrow 0} \frac{(1+x)^{1 / x}-e+\frac{1}{2} e x}{x^{2}}$
$\displaystyle\lim _{x \rightarrow 0} \frac{e\left(1-\frac{x}{2}+\frac{11}{24} x^{2}+\ldots\right)-e+\frac{1}{2} e x}{x^{2}}$
$=\frac{\frac{11}{24} e x^{2}+\ldots .}{x^{2}}=\frac{11}{24} e$