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Determine the value of a for a given function.

  1. Jan 28, 2013 #1
    Determine the value of "a" for a given function.

    1. The problem statement, all variables and given/known data

    Consider the following function: f(x) { ax+7, x<2 and 4x-a, x≥2

    A.) Determine the value of "a" such that lim x->2- (approaching from left) f(x) = 8. You mus demonstrate all limits.

    2. Relevant equations

    ax+7, x<2 and 4x-a, x≥2

    3. The attempt at a solution

    Approaching from left I got lim x->2- (2a+7) and from right lim x->2+ (8-a)
    Am I trying to find a? If I set both limits equal I get a = .33, but the question asks determine the value of such that lim x->2- f(x) = 8 ? That would mean a =1/2 because 2* 1/2 = 1 + 7 = 8, but I don't think that's right unless I'm reading the question wrong..


    Edit:
    Or do I just state that ax+7 since for this question says lim x->2- (approaching) left f(x) = 8 do I say coming from the left is 8 while from the right it's still 8-a then set those two equal to calculate for "a"? Getting -1 as my answer for "a"?
     
    Last edited: Jan 28, 2013
  2. jcsd
  3. Jan 28, 2013 #2

    Mark44

    Staff: Mentor

    Re: Determine the value of "a" for a given function.

    This is what you mean:
    $$ \lim_{x \to 2^-}f(x) = 2a + 7$$
    $$ \lim_{x \to 2^+}f(x) = 8 - a$$
    Yes, so that the left-sided limit as x approaches 2 equals 8.
    What are you saying here? 2 * 1/2 = 1 ≠ 8
     
  4. Jan 28, 2013 #3
    Re: Determine the value of "a" for a given function.

    Alright, then now I'm assuming set 8 = 8-a
     
  5. Jan 28, 2013 #4

    Mark44

    Staff: Mentor

    Re: Determine the value of "a" for a given function.

    No, for two reasons.

    1) You are supposed to
    2)If 8 = 8 - a, then a = -1 is NOT the solution.
     
  6. Jan 28, 2013 #5

    Mark44

    Staff: Mentor

    Re: Determine the value of "a" for a given function.

    For the left-side limit, a = 1/2.
    You had that, but wrote something that didn't make sense.

    If a = 1/2, then 2a + 7 = 2(1/2) + 7 = 8
     
  7. Jan 28, 2013 #6
    Re: Determine the value of "a" for a given function.

    Sorry, i realized that I switched it to a-8 for some odd reason. Nevermind, I don't know how I came up with that. My fault.

    8 = 8 - a
    a+8 = 8
    a = 0
    ?
     
  8. Jan 28, 2013 #7

    Mark44

    Staff: Mentor

    Re: Determine the value of "a" for a given function.

    The question is asking about the left-side limit, not the right-side limit.
     
  9. Jan 28, 2013 #8
    Re: Determine the value of "a" for a given function.

    Wait, but I thought the question said to determine the value of "a" such that lim x->2- f(x) = 8 meaning that approaching from the left lim x->2- = 8 and from the right that being 8-a, we solve for "a" which is from lim x->2+ (8-a) , so 8 = 8-a


    if it's the other way around a = 1/2
     
    Last edited: Jan 28, 2013
  10. Jan 28, 2013 #9
    Re: Determine the value of "a" for a given function.

    I think I get it.

    2a+7 = 8 then solve for a

    a = 1/2

    Thanks
     
  11. Jan 28, 2013 #10
    Re: Determine the value of "a" for a given function.

    I did say that, but I said it without solving so it was just an assumption by looking at the problem, the reason why I didn't go with it because it sounded too easy even though it made sense. My fault. Thank you.
     
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