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Math Help - Trig Limits

  1. #1
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    Trig Limits

    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.
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  2. #2
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    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)}
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  3. #3
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    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.
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  4. #4
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    \tan x-\cot x=-\frac{2}{\tan 2x}. Now put u=x-\frac\pi4 and work your trig. algebra.
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