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Homework Help: Find the values of a and b that make f continuous everywhere

  1. Sep 12, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the values of a and b that make f continuous everywhere. (Enter your answers as fractions.)


    2. Relevant equations
    1mLJ9.gif


    3. The attempt at a solution

    lim x->2 (x2-4)/(x-2) = 0

    lim x->2 4a - 2b +5
    lim x->3 9a - 3b +5

    lim x->3 12 - a + b


    4a - 2b + 5 = 0
    4a - 2b = -5

    9a - 3b +5 = 12 - a + b
    10a - 4b = 7


    4a - 2b = -5
    a = 2.5b/4

    10(2.5b/4) - 4b = 7
    6.25b - 4b = 7
    2.25b = 7
    b = 28/9


    I'm obviously missing something because I've been screwing around with this problem for well over an hour already and I haven't been able to solve it. I need to get this done by tomorrow morning so any help is appreciated.
     

    Attached Files:

  2. jcsd
  3. Sep 12, 2010 #2

    Dick

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    You've got the right method, but there's a problem with the first step. lim x->2 (x^2-4)/(x-2) isn't zero.
     
  4. Sep 12, 2010 #3
    I'm really not sure what to do if x->2 for the first equation is undefined...
     
  5. Sep 12, 2010 #4

    Dick

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    Try and factor the numerator and cancel the factor which is going to zero.
     
  6. Sep 12, 2010 #5
    (x+2)(x-2)/(x-2)
    (x+2)
    x->2 = 4

    That would mean that...

    4a - 2b + 5 = 4
    4a - 2b = -1
    4a = 1/2b
    a = 1/8b

    But plugging that into 10a - 4b = 7 gives me...

    10(1/8b) - 4b = 7
    (5/4)b - 4b = 7
    (-11/4)b = 7
    b = -28/11

    ...Okay, that isn't right...I did this problem 3 different times and came up with a new answer each time.
     
  7. Sep 12, 2010 #6

    Dick

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    You are goofing up your algebra. 4a-2b=(-1) doesn't lead to 4a=(1/2)b. What are you doing?
     
  8. Sep 12, 2010 #7
    I'm going to blame it on the embarrassingly long amount of time I've been looking at this problem. I just solved, finally. Thanks a lot! I really appreciate it!
     
  9. Sep 12, 2010 #8

    Dick

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    Staring at a problem for too long does make this stuff happen. Glad you got it.
     
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