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Math Help - Need some help Finding a limit (if it exists even)

  1. #1
    Junior Member
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    Need some help Finding a limit (if it exists even)

    Howdy,

    I need some help finding this limit:

    lim \Delta x -> 0

    \sqrt{x+\Delta x}-\sqrt{x}
    ______________________________

    \Delta x

    (P.S. Sorry I was having a bit of trouble using the \frac command, just so you know it's suppose to be delta X on the bottom of the fraction as I tried to show using the ___). Thanks.

    jimmyp
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by jimmyp View Post
    Howdy,

    I need some help finding this limit:

    lim \Delta x -> 0

    \sqrt{x+\Delta x}-\sqrt{x}
    ______________________________

    \Delta x

    (P.S. Sorry I was having a bit of trouble using the \frac command, just so you know it's suppose to be delta X on the bottom of the fraction as I tried to show using the ___). Thanks.

    jimmyp
    first we need to multiply the numerator and denominator by the conjugate of the numerator

    \lim_{\Delta x \to 0}\frac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x}\left( \frac{\sqrt{x+\Delta x}+\sqrt{x}}{\sqrt{x+\Delta x}+\sqrt{x}}\right)

    Simplifying we get

    \lim_{\Delta x \to 0}\frac{\Delta x}{\Delta x(\sqrt{x+\Delta x}+\sqrt{x})}=\lim_{\Delta x \to 0}\frac{1}{(\sqrt{x+\Delta x}+\sqrt{x})}

    Finally taking the limit we get

    \frac{1}{2\sqrt{x}}
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  3. #3
    MHF Contributor
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    Talking

    Quote Originally Posted by TheEmptySet View Post
    first we need to multiply the numerator and denominator by the conjugate of the numerator....
    That "multiplying by the conjugate" thing, by the way, is a "trick" you should remember. You will almost certainly need it on the next test!
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