154. Find $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{3 x}$.
Solution. We have

$$
\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{3 x}=\lim _{x \rightarrow \infty}\left(\left(1+\frac{2}{x}\right)^{x / 2}\right)^{6}=\left(\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x / 2}\right)^{6}
$$

Let $x / 2=y$. Then, as $x$ increases without bound, the variable $y$ will also increase without bound. Therefore, using the second remarkable limit, we get $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{x / 2}=\lim _{y \rightarrow \infty}\left(1+\frac{1}{y}\right)^{y}=$ $=e$. Thus, $\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{3 x}=e^{6}$.