Multivariable Limits, Squeeze Principle

[dr721]

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 lim_x→a f(x) = l = lim_x→a h(x). Prove that lim_x→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) - l ≤ g(x) - l ≤ h(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?

 

[voko]

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?

 

[dr721]

I guess I don't actually understand what epsilon and delta mean. Like, I'm struggling to understand how they define the limit.

 

[voko]

Do you understand how the limit is defined in the 1D case?

 

[dr721]

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.

 

[voko]

Start with this article, and come back if something is not clear: http://en.wikipedia.org/wiki/(ε,_δ)-definition_of_limit