easy way to calculate the limit $\lim_{x \to 0 } \frac{\cos{x}- (\cos{x})^{\cos{x}}}{1-\cos{x}+\log{\cos{x}}}$ I have been trying to use L'Hôpital over this, but its getting too long, is there a short and elegant solution for this?
The Limit approaches 2 according to wolfram.

Start with $$\cos(x)=1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6) \\
\log(\cos(x))=-\frac{x^2}{2}-\frac{x^4}{12}+O(x^6) \\
\cos(x)\log(\cos(x))=-\frac{x^2}{2}+\frac{x^4}{6}+O(x^6)$$
Then $$\cos(x)-e^{\cos(x)\log(\cos(x))}=1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6)-\left\{1+\cos(x)\log(\cos(x))+\frac{(\cos(x)\log(\cos(x)))^2}{2}+O\left((\cos(x)\log(\cos(x)))^3\right)\right\}\\
=-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6)-\left\{-\frac{x^2}{2}+\frac{x^4}{6}+O(x^6)+\frac{x^4}{8}+O(x^6)+O(x^6)\right\} \\
=-\frac{x^4}{4}+O(x^6) \,.$$
Similarly,
$$1-\cos(x)+\log(\cos(x))=1-\left\{1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6)\right\}+\left\{ -\frac{x^2}{2} - \frac{x^4}{12} + O(x^6) \right\}\\
=-\frac{x^4}{8}+O(x^6) \, .$$
