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Thread: Seeking limit of function with expression radicand in denominator

  1. #1
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    Seeking limit of function with expression radicand in denominator

    Hi all,

    I am in an early calculus chapter, where I am finding derivatives algebraically. (Hence why I am posting in pre-calculus; I have not yet covered techniques to differentiate functions efficiently.) I am asked to find the differential of the following function:

    {f}(x) = \frac{1}{\sqrt{x+2}}

    I am trying to find the derivative by using a limit function to determine instantaneous rate of change:

    {f}'(x) = \lim_{h \to 0} \left ( \frac{1}{h} \right ) \left (\frac{1}{\sqrt{x + h + 2}} -  \frac{1}{\sqrt{x + 2}} \right )

    However, I cannot solve this equation--I don't know how to get an 'h' factor in the numerator, which would allow me to cancel out the (1 / h) term. I have tried multiplying, dividing, and subtracting terms; I have tried multiplying the conjugates of radicals; I've been at this one for a while. Hopefully, one of you thinks this is obvious, and I've already over-explained a very simple problem... I've been staring at this one too hard, and I'm definitely missing the trick.

    Thank you!
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  2. #2
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    Re: Seeking limit of function with expression radicand in denominator

    Get a common denominator to combine the two terms. This gives:

    $\displaystyle \lim_{h\to 0} \dfrac{\sqrt{x+2}-\sqrt{x+h+2}}{h\sqrt{(x+h+2)(x+2)}}$

    Then multiply top and bottom by the conjugate.

    $\displaystyle \lim_{h\to 0}\dfrac{-h}{h\sqrt{(x+h+2)(x+2)}\left(\sqrt{x+2}+ \sqrt{x+h+2}\right)}$

    Can you solve it from there?
    Last edited by SlipEternal; Jun 22nd 2017 at 04:01 AM.
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  3. #3
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    Re: Seeking limit of function with expression radicand in denominator

    It looks so obvious when you do it. Thank you!
    Last edited by topsquark; Jun 22nd 2017 at 07:08 AM.
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