Concept preview

What you will be able to do

1. 01State the Squeeze (Sandwich) Theorem precisely, including the two conditions it requires.
2. 02Bound a hard function between two simpler functions that share a common limit, and conclude the trapped limit.
3. 03Apply the theorem to oscillating products such as $\lim_{x\to0}x^2\cos(1/x)=0$, and recognise when it does and does not apply.

Imagine a runner boxed in between two teammates roped together — one always on their left and one on their right. As the outer two funnel toward a single doorway, the runner in the middle is forced through that same doorway, whatever zig-zagging they were doing before. The Squeeze Theorem (also called the Sandwich or Pinching Theorem) is exactly this idea for functions: a function pinned between two others must go wherever those two others jointly go.