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Thread: evaluating a trig limit

  1. #1
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    evaluating a trig limit

    I am trying to evaluate lim of f(x)=COTx as x approaches $\displaystyle \frac{pi}{4}$ using the formula f'(a)=$\displaystyle \frac{cotx-cota}{x- a}$ and a=$\displaystyle \frac{pi}{4}}$
    so..
    $\displaystyle \frac{cotx-1}{x- \frac{pi}{4}}$

    The answer is is suppose to be -2. How can I find the steps to get to -2?
    I am sure I need to somehow use the trig limits $\displaystyle \frac{sinx}{x}=1$ and $\displaystyle \frac{cosx-1}{x}=0 $


    I start off like this, i suppose
    $\displaystyle \frac{\frac{cos}{sin}-1}{x- \frac{pi}{4}}$
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  2. #2
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    Re: evaluating a trig limit

    Certainly you mean something other than the limit... you would not need that formula, and that is not what the limit is equal to:
    lim cot(x) as x->pi/4 - Wolfram|Alpha

    Perhaps you mean the derivative of cotx? In that case, you want

    $\displaystyle \lim_{x\to\tfrac{\pi}{4}} \frac{\cot x - \cot \tfrac{\pi} 4} {x - \tfrac{\pi} 4}$

    which is what you have near the end there... I am just clarifying.

    Have you learned the derivative of sine and cosine yet?
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  3. #3
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    Re: evaluating a trig limit

    I believe the OP is trying to find $\displaystyle f'\left(\frac{\pi}{4}\right)$ using the limit process where $\displaystyle f(x) = \cot{x}$
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  4. #4
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    Re: evaluating a trig limit

    $\displaystyle f'(a) = \lim_{x \to a} \frac{\frac{\cos{x}}{\sin{x}} - \frac{\cos{a}}{\sin{a}}}{x-a}$

    $\displaystyle f'(a) = \lim_{x \to a} \frac{\frac{\cos{x}\sin{a} - \sin{x}\cos{a}}{\sin{x}\sin{a}}}{x-a}$

    difference identity for sine ...

    $\displaystyle f'(a) = \lim_{x \to a} \frac{\sin(a-x)}{(x-a)\sin{x}\sin{a}}$

    note that $\displaystyle \sin(a-x) = -\sin(x-a)$ since sine is an odd function...

    $\displaystyle f'(a) = \lim_{x \to a} -\frac{\sin(x-a)}{(x-a)\sin{x}\sin{a}}$

    $\displaystyle f'(a) = \lim_{x \to a} -\frac{\sin(x-a)}{x-a} \cdot \frac{1}{\sin{x}\sin{a}}$

    $\displaystyle f'(a) = \lim_{x \to a} -1 \cdot \frac{1}{\sin{x}\sin{a}} = -\frac{1}{\sin^2{a}}$
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