1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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


    User Avatar
    Science Advisor

    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


    User Avatar
    Science Advisor

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Left-handed limit of a rational function