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Limit problem

  1. Sep 25, 2007 #1
    1. The problem statement, all variables and given/known data


    Evaluate the limits in the terms of the constants involved:


    [itex]\lim_{x \rightarrow a} \frac{(x+a)^2-4x^2}{x-a}[/itex]
    2. Relevant equations



    3. The attempt at a solution

    First I plugged in a to make sure it was indeterminate and it was it ends up being 0/0.

    So I tried to break this up into the difference of two limits;


    [itex]\lim_{x \rightarrow a} \frac{(x+a)^2}{x-a}[/itex] -[itex]\lim_{x \rightarrow a} \frac{4x^2}{x-a}[/itex]

    but I again got stuck and I'm not even sure if this attempt is the correct way to go, I also tried to split the numerator into the difference of two squares but that lead to nothing, I'm not really sure where to go from here I'd appreciate it if someone could point me in the right direction.
     
  2. jcsd
  3. Sep 25, 2007 #2

    Hurkyl

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    Well, as the limit theorems say... if at least one of those two limits exist, then this is a valid way to compute the oriignal limit. Does either one exist?
     
  4. Sep 25, 2007 #3
    I would think they both do although they get 0 in the denominator they do not get 0 in the numerator nor infinity
     
  5. Sep 25, 2007 #4

    Hurkyl

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    Well, what do the limit theorems say about the limit of a fraction when the limit of the numerator is nonzero and the limit of the denominator is zero?
     
  6. Sep 25, 2007 #5

    Hurkyl

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    Hrm. I suppose it depends on how you are interpreting "exist". Certainly, the limit theorems tell you that the limit is not finite -- so if they exist, the only remaining possibilities are positive or negative infinity. So, there are very few possibilities; can you tell if these limits are +infinity, -infinity, or nonexistant?


    (My previous statements were made according to the interpretation that "doesn't exist" includes the csaes where the limit goes to +infinity or to -infinity)
     
  7. Sep 25, 2007 #6

    Hurkyl

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    Hrm. The limit theorems I remember include cases like:

    If [itex]\lim_{x \rightarrow a} f(x)[/itex] exists and is nonzero, and [itex]\lim_{x \rightarrow a} g(x) = 0[/itex], then [itex]\lim_{x \rightarrow a} f(x) / g(x)[/itex] is either infinite or nonexistant.​
     
  8. Sep 25, 2007 #7
    i know that if you had actual numbers instead of a I could tell if if they were going to infinity but with a variable its confusing me
     
  9. Sep 25, 2007 #8
    does it approach -infinity from the left but +infinity from the right won't the top stay 0 near the limit because near the limit x will be very close to the value of a and you will get 4a^2-4a^2
     
  10. Sep 25, 2007 #9
    waitttt hold on can't you use long division for these types of problems I am looking through my old calculus notebook and I see that I used long division for some
     
  11. Sep 25, 2007 #10

    Hurkyl

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    You're right on both counts! You have both determined why your approach won't work, and you have recalled an approach that will work!
     
  12. Sep 25, 2007 #11
    :_) is the answer -4a
     
  13. Sep 25, 2007 #12

    Hurkyl

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    I agree with that.
     
  14. Sep 25, 2007 #13
    thanks for all the help hurkyl
     
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