Question: I have to calculate this limit whitout using L'Hopital's rule or Taylor polynomials:

$$\lim\_{ x\to \pi/4 } \frac{1 - \tan(x)}{x-\frac{\pi}{4}}$$

I know how to make it using L'Hopital and that the result is $-2$ ,but I'm getting nowhere when I try without it. Any advice?

Answer: 
Hint: What is the definition of the derivative of $\tan(x)$ at $x=\pi/4$?
You could use the following hint. Try to write $\tan(x)$ in terms of $\sin(x) $ and $\cos(x)$. A little bit of rearranging and then use the following,

$$\lim\_{x\rightarrow 0} \frac{\sin(x)}{x}=1$$

A little bit of work yields the result.