I have the following limit:
$$\lim_{x\to \infty} \cos^{x^{2}}\left(\dfrac{a}{x}\right)$$
where $a$ can be any number.
How can I calculate it?
2026-04-06 13:45:37.1775483137
Evaluating $\lim_{x\to \infty} \cos^{x^{2}}\left(\frac{a}{x}\right)$
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$$\log\left(\cos^{x^2}(a/x)\right)= x^2\log\cos(a/x)=x^2\log\left(1-\frac {a^2}{2x^2}+O(x^{-4})\right) =x^2\left(-\frac {a^2}{2x^2}+O(x^{-4})\right)$$ so that $\cos^{x^2}(a/x)\to \exp(-a^2/2)$ as $x\to\infty$.