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Finding the Limit of a Rational Function

  1. Jul 12, 2011 #1
    This is just a problem I came across while reviewing basic calculus.

    1. The problem statement, all variables and given/known data

    Find the limit as x approaches 0 of f(x)=(1/(x(x+1)^1/2)) - (1/x)

    2. Relevant equations

    3. The attempt at a solution

    My problem here is really more of an algebra problem than a calculus problem. I cannot for the life of me remember how to get the limit of the denominator of either term to not equal 0. I've tried everything I can think of (rationalizing the denominator, etc.), to no avail.

    I could find no tutorials either online or in my book that use an example quite like this one. It's truly maddening, especially because I know I should have learned this 6 years ago in Algebra.

    Thanks in advance. Hope you can help.
  2. jcsd
  3. Jul 12, 2011 #2


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    Homework Helper

    Do you mean



    If yes, factor out 1/x and multiply and divide f(x) by [1/sqrt(x+1)+1].

  4. Jul 12, 2011 #3
    I think I must be misunderstanding what you're saying. Either that or I did the math wrong multiple times.

    Would you mind clarifying what exactly you mean? Thanks.
  5. Jul 13, 2011 #4


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    Homework Helper

    Use the notation 1/√(x+1)=a.

    f(x)=(1/x) (a-1)

    Multiply and divide by (a+1)



    Replace back 1/√(x+1) for a and simplify. Find the limit.
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