lim cos(1/n)^(n²)

Logarithm. $\ln\left(\cos\frac1n\right)^{n^2}=n^2\ln\cos\frac1n$.

Expansion. $\cos\frac1n=1-\frac1{2n^2}+o\!\left(\frac1{n^2}\right)$, and $\ln(1+v)=v+o(v)$, so
$$\ln\cos\frac1n=-\frac1{2n^2}+o\!\left(\frac1{n^2}\right).$$

Limit. $n^2\ln\cos\frac1n\to-\frac12$, hence the sequence converges to $e^{-1/2}$.

Remark. The exponent $n^2$ (and not $n$) is exactly what is needed to balance the $\frac1{n^2}$ deviation of the cosine from $1$. With exponent $n$, the limit would be $1$.