I need some help with this question.
lim x->pi/4 tan(x)-cot(x) / x - pi/4
Please go slow on me because we just started trig limits and i'm still quite confused. Any help would be great.
Recall that a derivative at a point is given by: $\displaystyle f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$
For this question, imagine $\displaystyle a = \frac{\pi}{4}$ and $\displaystyle f(x) = \tan x - \cot x$
Also note that: $\displaystyle f\left(\frac{\pi}{4}\right) = \tan \frac{\pi}{4} - \cot \frac{\pi}{4} = 0$
Putting it all together: $\displaystyle \lim_{x \to \frac{\pi}{4}} \frac{\overbrace{(\tan x - \cot x) - 0}^{f(x) - f(a)}}{\underbrace{x - \frac{\pi}{4}}_{x -a}} = \overbrace{\frac{d}{dx} \bigg|_{x = \frac{\pi}{4}} \left(\tan x - \cos x\right)}^{f'(a)}$