Example 4. Find $\lim _{x \rightarrow \infty}\left(\frac{x+2}{x-3}\right)^{x}$.

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Example 4. Find $\lim _{x \rightarrow \infty}\left(\frac{x+2}{x-3}\right)^{x}$.

The text above has been translated into English, please retain the original text's line breaks and format, and output the translation result directly.
Solution. Dividing the numerator and the denominator of the fraction by $x$ and using formula (1.64) for $k=+2$ and $k=-3$, we get

$$
\lim _{x \rightarrow \infty}\left(\frac{x+2}{x-3}\right)^{x}=\lim _{x \rightarrow \infty}\left(\frac{1+\frac{2}{x}}{1+\frac{-3}{x}}\right)^{x}=\frac{e^{2}}{e^{-3}}=e^{5}
$$