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Math Help - Evaluation of limit.

  1. #1
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    Evaluation of limit.

    The problem is to evaluate the limit using la'hospital rule..
    \lim_{x\rightarrow a} \left ( 2-\frac{x}{a} \right )^{tan\frac{\pi x}{2a}

    Soln.

    Let, y = \lim_{x\rightarrow a} \left ( 2-\frac{x}{a} \right )^{tan\frac{\pi x}{2a}}
    logy = \lim_{x\rightarrow a}  tan\frac{\pi x}{2a}\log \left (2-\frac{x}{a} \right )
    after applying La'Hospital rule, i gt this,
    logy =  \frac{\pi}{2} \lim_{x\rightarrow a} sec^{2}\frac{\pi x}{2a}.tan\frac{\pi x}{2a}

    ....i gt stuck in this step coz i gt repeatedly 0/0 form..n pliz suggest how do i write all these mathematical equations in faster and efficeint way, is there any software to make it possible???
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Evaluation of limit.

    You can rewrite:
    \lim_{x \to a} \left(2-\frac{x}{a}\right)^{\tan\left(\frac{\pi x}{2a}\right)}=e^{\lim_{x \to a}\frac{\ln\left(2-\frac{x}{a}\right)}{\tan\left(\frac{\pi x}{2a}\right)}

    And you can write \tan\left(\frac{\pi x}{2a}\right)=\frac{\sin\left(\frac{\pi x}{2a}\right)}{\cos\left(\frac{\pi x}{2a}\right)}

    Therefore you get:
    e^{\lim_{x \to a}\frac{\cos\left(\frac{\pi x}{2a}\right)\ln\left(2-\frac{x}{a}\right)}{\sin\left(\frac{\pi x}{2a}\right)}

    Can you compute the limit?
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  3. #3
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    Re: Evaluation of limit.

    m confuse how tan\frac{\pi x}{2a} comes in denominator...
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  4. #4
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    Re: Evaluation of limit.

    Quote Originally Posted by rishilaish View Post
    The problem is to evaluate the limit using la'hospital rule..
    \lim_{x\rightarrow a} \left ( 2-\frac{x}{a} \right )^{tan\frac{\pi x}{2a}

    Soln.

    log(y) = \lim_{x\rightarrow a}  tan\frac{\pi x}{2a}\log \left (2-\frac{x}{a} \right )
    after applying L'Hospital rule, i gt this,
    You can't apply L'H˘pital's Rule directly to a limit of the form (▒∞)(0) .
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  5. #5
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    Re: Evaluation of limit.

    Quote Originally Posted by SammyS View Post
    You can't apply L'H˘pital's Rule directly to a limit of the form (▒∞)(0) .
    No...i hv reduced it into 0/0 form and after applying l'hospital rule...still getting 0/0 form...
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  6. #6
    MHF Contributor Siron's Avatar
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    Re: Evaluation of limit.

    Sorry, my bad, it has to be:
    e^{\lim_{x \to a}\frac{\sin\left(\frac{\pi x}{2a}\right)\ln\left(2-\frac{x}{a}\right)}{\cos\left(\frac{\pi x}{2a}\right)}

    Now you can apply l'Hopital's rule.
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  7. #7
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    Re: Evaluation of limit.

    siron..pliz help me out wiht...Evaluation of limit(2)
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