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Math Help - evaluating limits

  1. #1
    Junior Member
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    evaluating limits

    If f(x) = sqrt(x+1) evaluate the following expression:

    lim (f(x+h)-f(x))/ h
    h->0

    so my equation looks something like:

    (sqrt(x+h+1)-sqrt(x+1))/h

    how do i simplify the square roots? I'm thinking the h's would cancel out somehow in order for this limit not to be undefined but i don't know how that would happen.
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Well you're actually contemplating a derivative via definition of derivative.

    Next step: multiply top & bottom by \sqrt {x + h + 1}  + \sqrt {x + 1} .
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  3. #3
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    <br />
(\sqrt {x + h + 1} - \sqrt {x + 1} )/ h.<br />

    the bottom is just h so how would i multiply the top by itself and the bottom by the numerator?
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  4. #4
    Super Member wingless's Avatar
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    \lim_{h\to 0} \frac{\sqrt{x+h+1}-\sqrt{x+1}}{h}

    \lim_{h\to 0} \frac{\sqrt{x+h+1}-\sqrt{x+1}}{h}\cdot \frac{\sqrt{x+h+1}+\sqrt{x+1}}{\sqrt{x+h+1}+\sqrt{  x+1}}

    \lim_{h\to 0} \frac{(x+h+1)-(x+1)}{h(\sqrt{x+h+1}+\sqrt{x+1})}

    \lim_{h\to 0} \frac{\not h}{\not h(\sqrt{x+h+1}+\sqrt{x+1})}

    \lim_{h\to 0} \frac{1}{\sqrt{x+h+1}+\sqrt{x+1}}

    Put h=0,

    \frac{1}{2\sqrt{x+1}}
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