Evaluate the limit

Problem:

\[ \lim_{x\to\infty}\left(1+\frac{2}{3x}\right)^{x} \]

Remark: another tiny justification is to set \(t=x\) and directly apply the identity, or write \(\left(1+\dfrac{2}{3x}\right)^{x}=\Big[\left(1+\dfrac{2}{3x}\right)^{3x/2}\Big]^{2/3}\) and use \(\lim_{u\to\infty}\big(1+\tfrac{1}{u}\big)^{u}=e\).