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Math Help - L'Hopital's with two variables

  1. #1
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    L'Hopital's with two variables

    Hi I'm having trouble solving the following limit:

    \lim_{t\to 0}\frac{t}{\sqrt(x + t) - \sqrt(x)}

    I've got as far as applying L'Hopital's rule to the top and bottom of the quotient, getting:
    <br />
\frac{\frac{dt}{dt}}{\frac{1 + \frac{dt}{dt}}{2\sqrt(x + t)}-\frac{1}{2\sqrt(x)}}

    As far as I know, this is the correct step to take. I'm clueless as to what I should do from here.
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  2. #2
    Senior Member roninpro's Avatar
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    This is simply the reciprocal of the difference quotient

    f'(x) = \displaystyle \lim_{t\to 0} \frac{f(x+t)-f(x)}{t}

    where \displaystyle f(x)=\sqrt{x}.

    Therefore, \displaystyle \lim_{t\to 0} \frac{t}{\sqrt{x+t}-\sqrt{x}}=\frac{1}{f'(x)}=2\sqrt{x}.
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  3. #3
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    Thanks for the response. Can it be done using L'Hopital's rule though? Because the question specifically states that I need to use the rule.
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by blackhug View Post
    Thanks for the response. Can it be done using L'Hopital's rule though? Because the question specifically states that I need to use the rule.
    Yes it can. You have the right idea but you took the derivative of the denominator wrong.

     \displaystyle \frac d{dt}\left(\sqrt{x+t}-\sqrt{x}\right) = \frac1{2\sqrt{x+t}} since we're treating  x as a constant.
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  5. #5
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    Also

    [LaTeX ERROR: Convert failed]

    taking the limit is now trivial!
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