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Math Help - Indeterminate limit, using l'hopitals

  1. #1
    Member nautica17's Avatar
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    Indeterminate limit, using l'hopitals

    I have to find the following:

    limit as x goes to 0 of x^sin(x)

    I know that I get an indeterminate form 0^0 but what do I do from there to find the actual limit? I know the answer should be equal to 1, but how do I set up the problem to get that? Do I use l'hopitals? And if so.. how?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by nautica17 View Post
    I have to find the following:

    limit as x goes to 0 of x^sin(x)

    I know that I get an indeterminate form 0^0 but what do I do from there to find the actual limit? I know the answer should be equal to 1, but how do I set up the problem to get that? Do I use l'hopitals? And if so.. how?
    Let L=\lim_{x\to0}x^{\sin x}.

    Then \ln L=\lim_{x\to 0}\frac{\ln x}{\csc x}\rightarrow \frac{\infty}{\infty}

    Now apply L'H˘pital's rule.

    Can you take it from here?
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  3. #3
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    Quote Originally Posted by nautica17 View Post
    I have to find the following:

    limit as x goes to 0 of x^sin(x)

    I know that I get an indeterminate form 0^0 but what do I do from there to find the actual limit? I know the answer should be equal to 1, but how do I set up the problem to get that? Do I use l'hopitals? And if so.. how?
    For questions similar to this, where the exponent plays a key role, it helps to consider an entirely different question, namely the logarithm of the limit

    So let's consider \lim_{x\rightarrow 0}\ln (x^{\sin(x)})=\lim_{x\rightarrow 0}\sin(x)\cdot \ln(x)=\lim_{x\rightarrow 0}\frac{\ln(x)}{\frac{1}{\sin(x)}}

    Now we can apply that L'hospital's rule to get

    \displaystyle \lim_{x\rightarrow 0}\frac{\frac{1}{x}}{\frac{-\cos(x)}{\sin^2(x)}}

    \lim_{x\rightarrow 0}\frac{-\sin^2(x)}{x\cos(x)}

    And via another LH Rule

    =\lim_{x\rightarrow 0}\frac{-2\sin(x)\cos(x)}{\cos(x)-x\sin(x)}

    =0

    So the ln of the limit=0 so what number do we plug into ln to get 0? The answer is 1

    There's gotta be a better way, but this is what I came up with....
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