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Math Help - Limit Defintion of the Derivative/ Rationalization help

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    Limit Defintion of the Derivative/ Rationalization help

    I have the function f(x)= the square root of(x+2). i must find the limit of x=2 as h approaches zero. When i rationalize the numerator in the problem i get (4h-4)/(2h times the square root of (h) +2h). I do not see anyway to simplify the problem. Can anyone see what im doing wrong/ not doing?
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  2. #2
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    Quote Originally Posted by gamefreeze View Post
    I have the function f(x)= the square root of(x+2). i must find the limit of x=2 as h approaches zero. When i rationalize the numerator in the problem i get (4h-4)/(2h times the square root of (h) +2h). I do not see anyway to simplify the problem. Can anyone see what im doing wrong/ not doing?
    \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}=

    \lim_{h \to 0}\frac{\sqrt{x+h+2}-\sqrt{x+h}}{h}=

    Now you want to multiply by the conjugate of the numerator to get

    \lim_{h \to 0}\frac{\sqrt{x+h+2}-\sqrt{x+h}}{h}\cdot \frac{\sqrt{x+h+2}+\sqrt{x+2}}{\sqrt{x+h+2}+\sqrt{  x+2}}=

    \lim_{h \to 0}\frac{h}{h(\sqrt{x+h+2}+\sqrt{x+2})}=\lim_{h \to 0}\frac{1}{\sqrt{x+h+2}+\sqrt{x+2}}=\frac{1}{2\sqr  t{x+2}}
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  3. #3
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    thank you very much. So it is easier to not input the values of x until i am done simplifying?
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  4. #4
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    Quote Originally Posted by gamefreeze View Post
    thank you very much. So it is easier to not input the values of x until i am done simplifying?
    It is a matter of opinion. I personally think so, but it can go either way. The reason I like to do it last is because it is more general, the formula is valid for any x, not just x=2.
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  5. #5
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    ok thanks a lot
    when not plugging in the values of x, the algebra is much easier to do.
    Last edited by gamefreeze; October 4th 2009 at 11:37 AM.
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