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Homework Help: Find the exact value of c for which f is continuous on its domain

  1. Mar 14, 2010 #1
    1. The problem statement, all variables and given/known data

    Let the function f: [1,infinity)-->R


    f(x)=\left\{ \begin{array}{rcl}
    \frac{(\sqrt{x}-1+x\sqrt{x-1}}{\sqrt{x^2-1}} & \mbox{,}
    & x>1 \\
    c, x=1

    Find the EXACT value of c for which f is continuous on its domain.

    2. Relevant equations


    3. The attempt at a solution

    I have sketched the graph on the computer and the first half (x greater than 1) part seems to be a single curve going to infinity on the right. I am not quite sure how to find c.

    Thanks in advance,
  2. jcsd
  3. Mar 14, 2010 #2


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    Science Advisor

    Well, can you state what it means for a function to be continuous?
  4. Mar 15, 2010 #3
    Ah, wait, continuous would mean that c would have to be the limit of the function as it tends towards x=1. So how do we evaluate the limit of this function? Can't seem to find a place to start...
  5. Mar 15, 2010 #4


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    Homework Helper

    Just curious: is the numerator

    \sqrt{x} -1 + x\sqrt{x-1}

    or is it

    \sqrt{x-1} + x\sqrt{x-1}

    It makes a difference in the attack and the answer.
  6. Mar 15, 2010 #5
    Hi, thanks for the reply. The numerator is indeed

    \sqrt{x} -1 + x\sqrt{x-1}

  7. Mar 15, 2010 #6


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    Science Advisor

    Try using L'Hopital's rule to evaluate the limit of your function as x->1.
  8. Mar 15, 2010 #7
    Oh yes, that brings up another point I forgot to mention. L'Hopital's rule is banned from this question. The only hint I got was to split the function into its sums...
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