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Multivariable Limits, Squeeze Principle

  • Thread starter dr721
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  • #1
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Homework Statement



(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?
 

Answers and Replies

  • #2
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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?
 
  • #3
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I guess I don't actually understand what epsilon and delta mean. Like, I'm struggling to understand how they define the limit.
 
  • #4
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Do you understand how the limit is defined in the 1D case?
 
  • #5
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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.
 

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