Evaluate $\lim \limits_{x \to 2} \frac{(\cos\theta)^x+(\sin\theta)^x-1}{x-2}, \theta \in (0,\frac{\pi}{2})$

Question

Evaluate $\lim \limits_{x \to 2} \frac{(\cos\theta)^x+(\sin\theta)^x-1}{x-2}, \theta \in (0,\frac{\pi}{2})$.

I've tried rationalizing, reducing it, etc. but I'm not getting anywhere. And by the way, I'm not allowed to use L'Hopital's rule.

Answer 1

HINT:

Write $1=\cos^2\theta+\sin^2\theta$

Now set $x-2=u$ in $$\lim_{x\to2}\dfrac{a^x-a^2}{x-2}=a^2\lim_{u\to0}\dfrac{a^u-1}u=a^2\ln a$$ using $\lim_{h\to0}\dfrac{e^h-1}h=1$ and $a=e^{\ln a}$