Evaluate the following Limits: _ x 0 (1+x 5 )^ 1 x - Vidyadip

Question

Evaluate the following Limits:

$\lim _{x \rightarrow 0}\left(1+\frac{x}{5}\right)^{\frac{1}{x}}$

✓

Answer

$\begin{aligned} & \lim _{x \rightarrow 0}\left(1+\frac{x}{5}\right)^{\frac{1}{x}} \\ & =\lim _{x \rightarrow 0}\left[\left(1+\frac{x}{5}\right)^{\frac{5}{x}}\right]^{\frac{1}{5}} \text { } \\ & =\mathrm{e}^{\frac{1}{5}} \quad \ldots\left[\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}=\mathrm{e}\right]\end{aligned}$

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