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Math Help - Limits of three functions

  1. #1
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    Limits of three functions

    Could you help me calculate the following limits:


    \lim_{x \to 0} x \left[  \frac{1}{x} \right]

    \lim_{x\to 0}  \frac{1-\cos x \cdot \sqrt{\cos2x} }{x^2}

    \lim_{x\to 10}  \frac{\log _{10}(x) - 1}{x-10}


    As to the last one I thought I could use \lim\frac{log _{a}(1+\alpha)}{\alpha} = \log_ae but it wouldn't work
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  2. #2
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    Re: Limits of three functions

    Is the first limit equal to 1, the second to 1,5 and the third one to 0,1\log_{10}e ?
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  3. #3
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    Re: Limits of three functions

    I just did the second one I think it's 3/2. I may have made some computational errors but you simply use l'Hopital's rule twice because you have the form 0/0 twice
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  4. #4
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    Re: Limits of three functions

    Quote Originally Posted by wilhelm View Post
    Is the first limit equal to 1, the second to 1,5 and the third one to 0,1\log_{10}e ?
    What an absolutely useless reply!
    Can you explain any of that? I doubt it.
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  5. #5
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    Re: Limits of three functions

    Limits of three functions-limit-q-24-dec.png
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  6. #6
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    Re: Limits of three functions

    @Plato But of course I can. To compute the first limit I simply used the squeeze theorem and floor function inequality. As to the second one I could use the de l'Hospital theorem but I used simple trigonometric transformations instead. And when it comes to the third one I just noticed that the limit is equal to derivative of log_{10}10.

    I posted the answers simply because I didn't feel like writing the whole solutions. I'm sorry.
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  7. #7
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    Re: Limits of three functions

    Quote Originally Posted by wilhelm View Post
    @Plato But of course I can. To compute the first limit I simply used the squeeze theorem and floor function inequality.
    It would have helped if you had said this problem involved the "floor function"! What you wrote, \left[\frac{1}{x}\right], is NOT the "floor function", it is just a complicated way of writing \frac{1}{x}.

    If you wanted the floor function, write it as \lfloor x \rfloor.

    As to the second one I could use the de l'Hospital theorem but I used simple trigonometric transformations instead. And when it comes to the third one I just noticed that the limit is equal to derivative of log_{10}10.

    I posted the answers simply because I didn't feel like writing the whole solutions. I'm sorry.
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