Mathematics - Limits Question with Solution | TestHub

Let $y=(1+x{)}^{1/x}$ $\Rightarrow \log y=\frac{1}{x}\log (1+x)=\frac{1}{x}\left[x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots \right]$ $=1-\frac{x}{2}+\frac{x^2}{3}-\frac{x^3}{4}+\dots$ $\begin{array}{rl}y& =e^{\left(1-\frac{x}{2}+\frac{x^2}{3}+\dots .\right)}=e{e}^{\left(\frac{-x}{2}+\frac{x^2}{3}\dots ..\right)}\\ & =e\left[1+\left(-\frac{x}{2}+\frac{x^2}{3}-\dots \right)+\frac{1}{2!}{\left(-\frac{x}{2}+\frac{x^2}{3}-\dots \right)}^2+\dots \right]\end{array}$ $y-e+\frac{1}{2}ex=e{x}^2\left[\frac{1}{3}+0(x)+\frac{1}{2}{\left(-\frac{1}{2}+0(x)\right)}^2+\dots .\right]$ $[\because 0(x)$ in terms containing $x]$ $\lim \frac{y-e+\frac{1}{2}ex}{x^2}=e\left[\frac{1}{3}+\frac{1}{8}\right]=\frac{11}{24}e$