calculus - Evaluate $lim_limits{x to 0} (1 +sin^2 x)^{frac{1}{ln(cos x)}}$

$$\lim_{x \to 0} (1 + \sin^2 x)^{\frac{1}{\ln(\cos x)}}$$

I evaluated $\sin$ and $\cos x$ but what can be done with $\ln\left(1-\frac{x^2}{2}\right)$ or $\ln\left(\frac{2 - x^2}{2}\right)$?

Assume that L'Hopital is forbidden but you can use asymptotic simplifications like big and small $o$ notations and Taylor series.

Answer

You can write the function as

$$(1 + \sin^2 x)^{ \frac{1}{\sin^2 x} \frac{\sin^2x}{\ln(\cos x)}}$$
Further

$$\frac{\sin^2x}{\ln(\cos x)}=\frac{x^2+o(x^2)}{\ln(1-\frac{x^2}{2}+o(x^2))}=\frac{x^2+o(x^2)}{-\frac{x^2}{2}+o(x^2)}\to-2$$

And

$$(1 + \sin^2 x)^{ \frac{1}{\sin^2 x} } \to e$$

Hence...