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Left-handed limit of a rational function

  1. Feb 21, 2015 #1
    1. The problem statement, all variables and given/known data
    What is Lim (1+x2)/(4-x) as x approaches 4 from the left? Prove using the definition.

    2. Relevant equations


    3. The attempt at a solution
    Well x≠4. Function approaches positive infinity as x approaches 4 from the left side. Let m>0 and 0<x<4.
    Then (1+x2)/(4-x) > x2/(4-x) > x/(4-x) > m when x...here I am stuck.
    So basically I need to show that if x>xm then (1+x2)/(4-x) >m?
    But at some point the same function approaches negative infinity so do a choose xm=(something, 4)?
     
    Last edited: Feb 21, 2015
  2. jcsd
  3. Feb 21, 2015 #2

    PeroK

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    It's usually a good idea to simlify things as much as possible before you start.

    What can you say about ##1+x^2## and ##1##?
     
  4. Feb 21, 2015 #3

    Svein

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    Well, (1 + x2)≥1 for all x. So for x<4, (1+x2)/(4-x)>1/(4-x). Now take an M>0. If (4-x)<1/M, then (1+x2)/(4-x)>M.
     
    Last edited: Feb 21, 2015
  5. Feb 21, 2015 #4
    When to use M and m? Does it matter?
     
  6. Feb 21, 2015 #5
    So 4 - 1/M < x < 4 ⇒(1+x2)/(4-x) > M >0 Q.E.D Thanks for super fast replies!
     
    Last edited: Feb 21, 2015
  7. Feb 21, 2015 #6

    HallsofIvy

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    As long you define M and m, it doesn't matter. In most textbook proofs, perhaps just because "M" is bigger than "m", "M" is use for a "maximum" or upper bound, "m" for a "minimum" or lower bound.
     
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