Evaluate: \lim_{x arrow+infinity} ((1+(2)/(x))^x)

Expression: $\lim_{x \rightarrow +\infty} \left({\left( 1+\frac{ 2 }{ x } \right)}^{x}\right)$

Substitute $t$ for $\frac{ 2 }{ x }$

$\lim_{t \rightarrow 0^+} \left({\left( 1+t \right)}^{\frac{ 2 }{ t }}\right)$

Use ${a}^{mn}={\left( {a}^{m} \right)}^{n}$ to transform the expression

$\lim_{t \rightarrow 0^+} \left({\left( {\left( 1+t \right)}^{\frac{ 1 }{ t }} \right)}^{2}\right)$

Use $\lim_{x \rightarrow c} \left({f\left( x \right)}^{a}\right)={\left( \lim_{x \rightarrow c} \left(f\left( x \right)\right) \right)}^{a}$ to transform the expression

${\left( \lim_{t \rightarrow 0^+} \left({\left( 1+t \right)}^{\frac{ 1 }{ t }}\right) \right)}^{2}$

Use $\lim_{x \rightarrow 0^+} \left({\left( 1+x \right)}^{\frac{ 1 }{ x }}\right)=e$ to evaluate the common limit

$\begin{align*}&{e}^{2} \\&\approx7.38906\end{align*}$