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Multivariable Limits, Squeeze Principle

  1. Sep 18, 2012 #1
    1. The problem statement, all variables and given/known data

    (Squeeze Principle) Suppose f, g, and h are real-valued functions on a neighborhood of a (perhaps not including the point a itself). Suppose f(x) ≤ g(x) ≤ h(x) for all x and limx→a f(x) = l = limx→a h(x). Prove that limx→a g(x) = l. (Hint: Given ε > 0, show that there is δ > 0 so that whenever 0 < ||x - a|| < δ, we have -ε < f(x) - lg(x) - lh(x) - l < ε.)

    2. The attempt at a solution

    I don't understand the definition of a limit with ε and δ. The question confuses me, frankly, and I don't have any idea where I would begin. Could anyone help me understand this?
     
  2. jcsd
  3. Sep 19, 2012 #2
    There are two equivalent definitions of the limit, in terms of sequences and in terms of epsilon-delta. Both can be used to prove the squeeze principle.

    What is your difficulty with the epsilon-delta definition?
     
  4. Sep 19, 2012 #3
    I guess I don't actually understand what epsilon and delta mean. Like, I'm struggling to understand how they define the limit.
     
  5. Sep 19, 2012 #4
    Do you understand how the limit is defined in the 1D case?
     
  6. Sep 19, 2012 #5
    Actually, I don't know that specific definition at all. My calculus teacher spent very little time doing limits. We learned the basic skill of taking a limit and L'Hopital's Rule, and then went straight into derivatives.

    I know a limit is a way of looking at the continuity/discontinuity of a function, but that's about it.
     
  7. Sep 19, 2012 #6
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