$\lim_{t \rightarrow 0^+} \left({\left( 1+t \right)}^{\frac{ 2 }{ t }}\right)$
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$\begin{align*}&{e}^{2} \\&\approx7.38906\end{align*}$