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Homework Help: I am stuck on evaluating the limit of a rational function

  1. Mar 8, 2010 #1
    1. The problem statement, all variables and given/known data

    Evaluate the limit: [tex]lim_{(x)\rightarrow(2)}\frac{x^2+x-6}{sqrt(x+4)-sqrt(6)}[/tex]

    2. Relevant equations

    N/A

    3. The attempt at a solution

    I've drawn the graph which indicates that the at x=2, y=0, so 0 would seem to be the limit.
    I could not, however, get the limit expression by algebraic manipulation to get to 0.

    First of all, I rationalized the denominator to give x-2, which is a common factor of the numerator and the denominator. But canceling this, I am left with the expression:

    [tex]lim_{(x)\rightarrow(2)}((x+3)(sqrt(x+4)+sqrt(6))[/tex]

    after the [tex]lim_{(x)\rightarrow(2)}\frac{x-2}{x-2}[/tex] comes to 1.

    I may have missed something crucial, so I just can't get the limit to be 0

    Any help is greatly appreciated,
    Charismaztex
     
  2. jcsd
  3. Mar 8, 2010 #2

    statdad

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    Try again - the limit is not zero. Your simplification is correct.
     
  4. Mar 8, 2010 #3

    HallsofIvy

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    That is NOT, by the way, a rational function.
     
  5. Mar 8, 2010 #4
    @statdad The graph shows that at x=2, y=0. How is this explained?

    @ HallsofIvy Why not? I thought it can be expressed in the form p/q where p and q are real. Or if there are irrational numbers anywhere, does this make it void?
     
  6. Mar 8, 2010 #5

    Hurkyl

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    A rational function is a ratio of polynomials.
     
  7. Mar 8, 2010 #6

    statdad

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    Check your graph again. the one i've attached doesn't show that: in fact, there is no point on the graph at x = 2 because of the denominator.
    A rational number is a ratio p/q where both numerator and denominator are integers. A rational expression is a fraction in which both numerator and denominator are polynomials.
     

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