The value of 78.6$\sqrt{e}\underset{x\to 0}{\lim }{\left\{\frac{{\left(1+x\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}}}{e}\right\}}^{\frac{1}{x}} is$

$\text{ Let }\mathrm{P}=\underset{\mathrm{x}\to 0}{lim} {\left\{\frac{(1+\mathrm{x}{)}^{1/\mathrm{x}}}{\mathrm{e}}\right\}}^{1/\mathrm{x}}$

$\begin{array}{l}={\mathrm{e}}^{\underset{x\to 0}{\lim }\left(\frac{{\left(1+\mathrm{x}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}}}{\mathrm{e}}-1\right)\frac{1}{\mathrm{x}}}\\ ={\mathrm{e}}^{\frac{1}{\mathrm{e}}\underset{\mathrm{x}\to 0}{lim} \frac{(1+\mathrm{x}{)}^{1/\mathrm{x}}-\mathrm{e}}{\mathrm{x}} } \\ \\ ={\mathrm{e}}^{\frac{1}{\mathrm{e}}(-\mathrm{e}/2)}={\mathrm{e}}^{-1/2}=\frac{1}{\sqrt{\mathrm{e}}} \left(\sin ce \underset{\mathrm{x}\to 0}{lim} \frac{(1+\mathrm{x}{)}^{1/\mathrm{x}}-\mathrm{e}}{\mathrm{x}}=\underset{\mathrm{x}\to 0}{lim}\frac{e\left(1-{\displaystyle \frac{x}{2}}+{\displaystyle \frac{11}{24}}{x}^2..\right)-e}{x} =\frac{-e}{2}\right) \end{array}$

$\therefore 78.6\sqrt{\mathrm{e}}\underset{\mathrm{x}\to 0}{lim} {\left\{\frac{(1+\mathrm{x}{)}^{1/\mathrm{x}}}{\mathrm{e}}\right\}}^{1/\mathrm{x}}=78.6\sqrt{\mathrm{e}}\cdot \frac{1}{\sqrt{\mathrm{e}}}=78.6$