A question about a limit

Question

I have the next $\lim_{x \to \frac{\pi}{4}} \frac{f(x)-f(\frac{\pi}{4})}{x-\frac{\pi}{4}}$ when the function is $$f(x)=\frac{x+\sin(x)}{\tan(x)}.$$ I don't know how to even start. I am sorry that this is short but I really don't know.

Answer 1

$f$ is said to be differentiable at $c$, if there exists a real number $\alpha$ such that $$f'(c)=\lim_{x\to c}\frac{f(x)-f(c)}{x-c}=\alpha.$$ Now see your question.

Find $f'$ and then put $x=\frac{π}{4}.$

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Note that $\frac{d}{dx}f=f'.$ If $$f(x)=\frac{h(x)}{g(x)}$$, then $$f'(x)=\frac{g(x)\frac{d}{dx}h(x)-h(x)\frac{d}{dx}g(x)}{(g(x))^2}.$$