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Limit with radical problem

  1. Jan 18, 2010 #1
    1. The problem statement, all variables and given/known data

    lim as h->0 of (f(x(sub 0)+h)-f(x(sub 0))/h

    f(x)=3[tex]\sqrt{x}[/tex]+2
    x(sub 0)=9

    2. Relevant equations

    limit laws and factoring (my first post, not sure what I need to write here)


    3. The attempt at a solution

    =lim as h->0 of (f(9+h)-f(9)0/h
    =lim as h->0 of (3[tex]\sqrt{9+h}[/tex]+2-(3[tex]\sqrt{9}[/tex]+2)/h
    =lim as h->0 of (3[tex]\sqrt{9+h}[/tex]+2-3[tex]\sqrt{9}[/tex]-2)/h
    =lim as h->0 of (3[tex]\sqrt{9+h}[/tex]-9)/h

    I am stuck here. I know I need to somehow move h out of the denominator(most likely by factoring) but am stuck on how to deal with the radical in the numerator.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 18, 2010 #2

    vela

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    Rationalize the numerator by multiplying both the top and bottom by [tex]3\sqrt{9+h}+9[/tex].
     
  4. Jan 18, 2010 #3
    Ok, so I multiplied the numerator and denominator by 3[tex]\sqrt{9+h}[/tex]+9 and am still having problems, here's what I got:

    =lim as h->0 of (3[tex]\sqrt{9+h}[/tex]-9)/h * (3[tex]\sqrt{9+h}[/tex]+9)/(3[tex]\sqrt{9+h}[/tex]+9)

    =lim as h->0 of (9(9+h)+27[tex]\sqrt{9+h}[/tex]-27[tex]\sqrt{9+h}[/tex]-81)/(3h[tex]\sqrt{9+h}[/tex]+9h)

    =lim as h->0 of (81+h-81)/(3h[tex]\sqrt{9+h}[/tex]+9h)

    =lim as h->0 of h/(3h[tex]\sqrt{9+h}[/tex]+9h)

    ok, I am stuck here again and don't see what I did wrong or what I still need to do
     
  5. Jan 18, 2010 #4
    You didn't distribute the 9 in the second line of your work; that will give you 9h in the numerator. Also, don't distribute the h in the denominator.
    This is what you should have now without distributing it:

    [tex]\lim_{h\rightarrow 0} \frac{9h}{h(3\sqrt{9 + h} + 9)}[/tex]

    The h's cancel and then you can let h=0 in the expression.
     
    Last edited: Jan 18, 2010
  6. Jan 18, 2010 #5
    Thanks for the help vela, you got me on the right track, and thanks Bohrok, you caught my mistake. I started over and worked it out and got the right answer.
     
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