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Having trouble evaluating these limits

  1. Nov 1, 2011 #1
    1. The problem statement, all variables and given/known data
    The limit as x approaches 2^+ of (x-4)/(x-2)

    The limit as x approaches 2^- of (x-4)/(x-2)


    2. Relevant equations



    3. The attempt at a solution
    Figured out that the domain is x ≠ 2, therefore 2 is a vertical asymptote
    not sure where to input values/what to do really..
     
  2. jcsd
  3. Nov 1, 2011 #2
    drawing a picture will help, i think.
     
  4. Nov 1, 2011 #3
    I tried
    lim
    x→2+ (0+-4)/(0+-2)
    but i keep getting + infinity

    and i'm not even sure if that's how you approach this question.

    Can anyone help?
     
    Last edited: Nov 1, 2011
  5. Nov 1, 2011 #4

    Mark44

    Staff: Mentor

    Did you follow murmillo's suggestion?

    As x approaches 2 from the right, what does the graph of y = (x - 4)/(x - 2) do?
    As x approaches 2 from the left, what does the graph of y = (x - 4)/(x - 2) do?
     
  6. Nov 1, 2011 #5
    I did not because I want to follow the method that my instructor used
    i did however draw the asymptote
     
  7. Nov 1, 2011 #6

    SammyS

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    And what method is that?
     
  8. Nov 1, 2011 #7
    That's what i'm trying to piece together.
    there was something about 1. (+)/(0^-) = - infinity
    2. (-)/(0^-1) = + infinity
    3. (-2)/(0^+)= - infinity
    4. (1)/(0^+) = infinity

    it has worked so far for my previous examples such as:
    lim
    x→1+ (3/x-1)

    Domain: x can not equal 1
    V.A: x = 1

    lim (3)/(0+-1) = + infinity

    but it is not working for the example that i'm on and i don't know why
     
  9. Nov 1, 2011 #8

    SammyS

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    For [itex]\displaystyle \lim_{x\to2^+} \frac{x-4}{x-2}[/itex]
    As x approaches 2 from the right (from the positive x direction) x-2 → 0+, that is to say x-2 becomes very small, but is positive.

    Also, as x approaches 2 from the right, x-4 is close to positive 2.

    So this gives you (+2)/(0+) → +∞​

    The limit as x → 2- can be handled similarly.
     
  10. Nov 1, 2011 #9
    however, for the first limit, the answer in the back of the book comes up as -∞
    and when i type the limit as x approaches 2^+ of (x-4)/(x-2) into wolfram alpha it also comes up as - infinity
    http://www.wolframalpha.com/input/?i=the+limit+as+x+approaches+2^++of+(x-4)/(x-2)
     
  11. Nov 1, 2011 #10
    can anyone else help?
     
  12. Nov 1, 2011 #11

    SammyS

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    DUH !!

    I made a mistake !

    ... , as x approaches 2 from the right, x-4 is close to -2 (negative 2).

    Therefore, this gives you (-2)/(0+) → -∞
     
  13. Nov 1, 2011 #12
    sorry, but i'm having trouble understanding why x-4 is close to -2
     
  14. Nov 2, 2011 #13

    Mark44

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    The whole statement is, when x is close to 2, x - 4 is close to -2.
     
  15. Nov 2, 2011 #14

    Mark44

    Staff: Mentor

    You should still sketch a graph of the function. The intent wasn't meant to be a proof, but to enable you to have some insight as to what the limit should be.
     
  16. Nov 2, 2011 #15
    finally got the concept!
    thanks everyone :)
     
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