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Find an upper bound to the limit of a function

  1. Oct 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Find an upper bound M for f(x) = |x-2 / x+(1/2)| if |x+1| < 1/4


    2. Relevant equations



    3. The attempt at a solution

    I'm confused about this |x+1| < 1/4. Does this mean that |x-1| < 1/4?

    |x-2/x+(1/2)| = x-2/(2x+1)/2 = 2(x-2)/(2x+1) = 2x - 4/2x + 1 = x-2/x+(1/2) < M

    Given -1/4 < |x+1| < 1/4
    -5/4 < x < -3/4
    -13/4 < x - 2 < -11/4

    Also
    -3/4 < x + (1/2) < -1/4
    then
    -4 < 1/ x + (1/2) < -4/3
    3 < x - 2 / x + (1/2) < 1/3 ???
     
  2. jcsd
  3. Oct 12, 2009 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    "I'm confused about this |x+1| < 1/4. Does this mean that |x-1| < 1/4?"

    No. Why would it mean that? You have an (x + 1/2) in the denominator. (At least I suppose you do; it should have parentheses around it.) So you don't want x to get too close to -1/2. Doesn't |x+1| < 1/4 help you with that?
     
  4. Oct 12, 2009 #3
    If |x+1| < 1/4
    -1/4 < x+1 < 1/4
    -5/4 < x < -3/4
    x cannot = -1/2

    What does "upper bound M" mean?
     
  5. Oct 12, 2009 #4

    Mark44

    Staff: Mentor

    A number M such that f(x) <= M for all x in some set. For example, M = 1 is an upper bound for g(x) = sin(x), for all real numbers x. Also, 2 is an upper bound, as is 1.7. 1 is the least upper bound.
     
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