Trig Limits

**Shapeshift** · November 23rd 2008, 05:33 PM

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.

**o_O** · November 23rd 2008, 05:40 PM

Recall that a derivative at a point is given by: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

For this question, imagine $a = \frac{\pi}{4}$ and $f(x) = \tan x - \cot x$

Also note that: $f\left(\frac{\pi}{4}\right) = \tan \frac{\pi}{4} - \cot \frac{\pi}{4} = 0$

Putting it all together: $\lim_{x \to \frac{\pi}{4}} \frac{\overbrace{(\tan x - \cot x) - 0}^{f(x) - f(a)}}{\underbrace{x - \frac{\pi}{4}}_{x -a}} = \overbrace{\frac{d}{dx} \bigg|_{x = \frac{\pi}{4}} \left(\tan x - \cos x\right)}^{f'(a)}$

**Shapeshift** · November 23rd 2008, 06:09 PM

Thanks for the reply.

But I don't quite understand it. I'm really new at this, so do you think you could explain it to me step by step? It would be great if you could.

**Krizalid** · November 24th 2008, 06:34 PM

$\tan x-\cot x=-\frac{2}{\tan 2x}.$ Now put $u=x-\frac\pi4$ and work your trig. algebra.

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