6.60. $\lim _{x \rightarrow+0}(\cos x)^{1 / x^{2}}$.

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6.60. $\lim _{x \rightarrow+0}(\cos x)^{1 / x^{2}}$.
Solution.

$$
\begin{aligned}
& \lim _{x \rightarrow+0}(\cos x)^{1 / x^{2}}=\left(1^{\infty}\right)=e^{\lim _{x \rightarrow 0} \frac{\ln \cos x}{x^{2}}}, \\
& \lim _{x \rightarrow+0} \frac{\ln \cos x}{x^{2}}=\lim _{x \rightarrow+0} \frac{(\ln \cos x)^{\prime}}{\left(x^{2}\right)^{\prime}}=\lim _{x \rightarrow+0} \frac{-\frac{\sin x}{\cos x}}{2 x}=-\frac{1}{2} \\
& \lim _{x \rightarrow+0}(\cos x)^{1 / x^{2}}=e^{-1 / 2}
\end{aligned}
$$