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Existence of limit of a function with a parameter

  1. Jun 14, 2014 #1
    1. The problem statement, all variables and given/known data
    For what values of a, from the reals, does the limit exist?
    [itex]lim_{x\rightarrow2} (\frac{1}{2-x}-\frac{a}{4-x^{2}})[/itex]

    2. Relevant equations
    I chose a so that the denominator would be one. By putting the fractions together.

    3. The attempt at a solution
    When a = 4 the denominator of the combined fraction can be reduced to one
    => then the limit is -1/4.

    (For [itex]a=x+2[/itex] and [itex]a=x^{2}+x-2[/itex] the denominator is 1 too, but at x=2 all three solutions are equal to 4.)

    [itex]\textbf{My question: Is 4 the only solution?}[/itex]

    In the problem statement a is in plural.
    Am I missing any solutions?

    Any hints are much appreciated.
  2. jcsd
  3. Jun 14, 2014 #2


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    Science Advisor
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    Gold Member

    You have the correct answer. To see that there are no other answers, add the two fractions together and see if you can argue that having a finite limit implies a = 4.
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