# A question about a limit

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.

• Take a look at that limit. Look at the form of it. Does it look, in any way, shape or form, familiar? Commented Feb 7, 2021 at 11:57
• What is the definition of the derivative of a function? Any relation with your limit? Commented Feb 7, 2021 at 11:57
• @Arthur I can't see, can you show me? Commented Feb 7, 2021 at 11:59
• Give it a try!! What is the definition of the derivative of a function? Commented Feb 7, 2021 at 12:00
• You have a BIG hint. Read en.wikipedia.org/wiki/Derivative#Rigorous_definition Now it is your turn! Commented Feb 7, 2021 at 12:06

$$f$$ 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 $$x=\frac{π}{4}.$$
Note that $$\frac{d}{dx}f=f'.$$ If $$f(x)=\frac{h(x)}{g(x)}$$, then $$f'(x)=\frac{g(x)\frac{d}{dx}h(x)-h(x)\frac{d}{dx}g(x)}{(g(x))^2}.$$
• so is it $1-\frac{\pi +\sqrt{2}}{2}$ Commented Feb 7, 2021 at 12:16