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Math Help - limit

  1. #1
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    limit

    find limit x->pi [(x-pi).cotx].

    I am stuck at x-pi/tanx
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: limit

    Have you tried L'H˘pital's rule?
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  3. #3
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    Re: limit

    no. But I found the answer.
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  4. #4
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    Re: limit

    tanx=tan(x-pi) and this makes 1.
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  5. #5
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    Re: limit

    Quote Originally Posted by kastamonu View Post
    find limit x->pi [(x-pi).cotx].

    I am stuck at x-pi/tanx
    \displaystyle \begin{align*} \lim_{x \to \pi} \left( x - \pi \right) \cot{(x)} &= \lim_{x \to \pi} \frac{\left( x - \pi \right) \cos{(x)}}{\sin{(x)}} \\ &= \lim_{x \to \pi} \frac{- \left( \pi - x \right) \left[ -\cos{ \left( \pi - x \right) }  \right] }{\sin{ \left( \pi - x \right)}} \\ &= \lim_{x \to \pi} \left[ \frac{\pi - x}{\sin{ \left( \pi - x \right) } } \right] \lim_{x \to \pi} \cos{ \left( \pi - x \right) } \\ &= 1 \cdot 1 \\ &= 1   \end{align*}
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  6. #6
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    Re: limit

    Many Thanks. I found it by using tanx/x=1
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  7. #7
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    Re: limit

    `tan (a - b)` = `(tan a - tan b)/(1 + tan a*tan b)`

    `tan (x - pi )` = `(tan x - tan pi)/(1 + tan x*tan pi)`

    tan pi = sin pi/cos pi = 0/-1 = 0`

    `tan (x - pi) = (tan x - 0)/(1 + tan x*0)`

    `tan(x - pi) = tan x`
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