Cosine to the power 1/x²

Idea. Another "$1^\infty$" form: the base tends to $1$ and the exponent to $+\infty$. We take the logarithm to turn the power into a product.

Computation. For $x$ near $0$, $\cos x>0$ and
$$(\cos x)^{1/x^2}=\exp\!\left(\frac{\ln\cos x}{x^2}\right).$$
Write $\cos x=1+h$ with $h=-\frac{x^2}2+o(x^2)$. Then $h\to0$ and
$$\ln\cos x=\ln(1+h)=h+o(h)=-\frac{x^2}2+o(x^2).$$
Hence $\dfrac{\ln\cos x}{x^2}\to-\dfrac12$, and by continuity of $\exp$ the limit is $e^{-1/2}=\dfrac1{\sqrt e}$.

Remark. The pattern is general: if $u(x)\to1$, then $u(x)^{1/x^2}$ is governed by the limit of $\dfrac{u(x)-1}{x^2}$.