Trapped in the Middle
If a function \(f(x)\) is always trapped between two other functions \(g(x)\) and \(h(x)\), and both of those functions are going to the same limit \(L\), then \(f(x)\) must also go to \(L\). This is the Squeeze Theorem (or Sandwich Theorem).
Worked Examples
Example 1: The Oscillating Zero
Find \(\lim_{x \to 0} [x^2 \sin(1/x)]\).
- We know \(\sin(1/x)\) bounces between -1 and 1 forever.
- So the whole function is trapped between \(-x^2\) and \(x^2\).
- As \(x \to 0\), both \(-x^2\) and \(x^2\) go to 0.
- Result: The limit is 0.
The Bridge to Quantum Mechanics
This theorem is used to prove the most famous limit in physics: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\). This limit is the reason why, for small angles, a wave behaves like a straight line. In Quantum Field Theory, we often deal with waves that have infinite "wiggles" near a certain point. We use the Squeeze Theorem to prove that these wiggles don't cause the total probability to blow up to infinity. It is a tool for proving that even the most chaotic-looking quantum systems are actually well-behaved and predictable.