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.

$$\lim_{x \to 0 } \frac{\cos(x)- (\cos(x))^{\cos(x)}}{1-\cos(x)+\log{\cos(
x)}}=\lim_{t \to 1 }\frac{t-t^t}{1-t+\log (t)}$$
$$t^t=1+(t-1)+(t-1)^2+\frac{1}{2} (t-1)^3+O\left((t-1)^4\right)$$
$$\log(t)=(t-1)-\frac{1}{2} (t-1)^2+\frac{1}{3} (t-1)^3+O\left((t-1)^4\right)$$
$$\frac{t-t^t}{1-t+\log (t)}=\frac{-(t-1)^2-\frac{1}{2} (t-1)^3+O\left((t-1)^4\right)} {-\frac{1}{2} (t-1)^2+\frac{1}{3} (t-1)^3+O\left((t-1)^4\right) }$$
$$\frac{t-t^t}{1-t+\log (t)}=2+\frac{7 }{3}(t-1)+O\left((t-1)^2\right)$$
