Example 11. Find $\lim _{x \rightarrow \infty}\left(\frac{2 x-1}{3 x+4}\right)^{x^{2}}$.
Solution. This is also a limit of the form $(1.60)$, where $\varphi(x)=\frac{2 x-1}{3 x+4}, \psi(x)=x^{2}$

Since

$$
\lim _{x \rightarrow \infty} \frac{2 x-1}{3 x+4}=\lim _{x \rightarrow \infty} \frac{2-\frac{1}{x}}{3+\frac{4}{x}}=\frac{2}{3}, \lim _{x \rightarrow \infty} x^{2}=\infty
$$

then according to formula (1.55)

$$
\lim _{x \rightarrow \infty}\left(\frac{2 x-1}{3 x+4}\right)^{x^{2}}=0
$$