Example 1. Find $\lim _{x \rightarrow \infty}\left(1+\frac{k}{x}\right)^{x}$.
Solution. When $\quad x \rightarrow \infty$, the expression $\quad\left(1+\frac{k}{x}\right)^{x} \rightarrow 1$, resulting in the indeterminate form $1^{\infty}$.

Introduce a new variable $\alpha$ by the formula

$$
\frac{k}{x}=\alpha
$$

from which $x=\frac{k}{\alpha}$. If $x \rightarrow \infty$, then $\alpha \rightarrow 0$, therefore

$$
\lim _{x \rightarrow \infty}\left(1+\frac{k}{x}\right)^{x}=\lim _{\alpha \rightarrow 0}(1+\alpha)^{\frac{k}{\alpha}}=\lim _{\alpha \rightarrow 0}\left[(1+\alpha)^{\frac{1}{\alpha}}\right]^{k}
$$

Using property (1.45), based on formula (1.58), we find

$$
\lim _{\alpha \rightarrow 0}\left[(1+\alpha)^{\frac{1}{\alpha}}\right]^{k}=\left[\lim _{\alpha \rightarrow 0}(1+\alpha)^{\frac{1}{\alpha}}\right]^{k}=e^{k}
$$

Therefore,

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

In particular, if $k=3$, then

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

if $k=-2$, we get

$$
\lim _{x \rightarrow \infty}\left(1-\frac{2}{x}\right)^{x}=e^{-2}
$$