How do you find the limit of (1 + (1/x))^(x/6) as x approaches infinity?

May 22, 2016

${e}^{\frac{1}{6}}$

Explanation:

${\lim}_{x \rightarrow \infty} {\left(1 + \frac{1}{x}\right)}^{x} = e$

and

${\left({1}_{1} / x\right)}^{\frac{x}{6}} = {\left({\left(1 + \frac{1}{x}\right)}^{x}\right)}^{\frac{1}{6}}$

So,

${\lim}_{x \rightarrow \infty} {\left({1}_{1} / x\right)}^{\frac{x}{6}} = {\lim}_{x \rightarrow \infty} {\left({\left(1 + \frac{1}{x}\right)}^{x}\right)}^{\frac{1}{6}}$

$= {\left({\lim}_{x \rightarrow \infty} {\left(1 + \frac{1}{x}\right)}^{x}\right)}^{\frac{1}{6}}$

$= {e}^{\frac{1}{6}}$
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