Question

Evaluate the limit ${lim}_{x\to 0}2(\frac{\ell n(1+x{)}^{1+x}}{x^2}-\frac{1}{x})$

• A. $1$
• B. $2$
• C. $3$
• D. $Noneofthese$

Answer 1

The correct option is A $1$

Now,

$\begin{array}{c}=2\times \left[{lim}_{x\to 0}\frac{\left(1+x\right)\ln \left(1+x\right)-x}{x^2}\right]\\ =2\times \left[{lim}_{x\to 0}\frac{\left(1+x\right)\left(x-\frac{x^2}{2}\right)-x}{x^2}\right]\\ =2\times \left[{lim}_{x\to 0}\frac{x+x^2-\frac{x^2}{2}-x}{x^2}\right]\\ =2\times \frac{1}{2}=1\\ hence,weget\\ =1\end{array}$

so option $A$ is correct answer.