# 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 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?

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.