Squeeze Theorem

The Squeeze Theorem

The squeeze theorem (also called the sandwich theorem) is a powerful tool for evaluating limits when direct substitution or algebraic manipulation is difficult. It allows us to "squeeze" a function between two others whose limits we know.

Statement of the Theorem

If g(x) ≤ f(x) ≤ h(x) for all x near a (except possibly at a), and:

The function f(x) is "squeezed" between g(x) and h(x), both of which approach the same limit L, so f(x) must also approach L.

Geometric Interpretation

Think of f(x) as being trapped between two "walls" g(x) and h(x). If both walls approach the same height L as x approaches a, then f(x) must also approach that height—it's squeezed to the same limit.

Classic Example: \(\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right)\)

The function \(\sin(1/x)\) oscillates wildly as x → 0, but it's always bounded between -1 and 1:

Multiplying by x² (which is positive for small x):

As x → 0, both -x² and x² approach 0:

Applications to Trigonometry

The squeeze theorem is essential for proving the fundamental limit:

Key Requirements

1. g(x) ≤ f(x) ≤ h(x) must hold near a (except possibly at a)
2. Both g(x) and h(x) must have the same limit L at a
3. The inequality doesn't need to hold at a itself, only near it

This theorem is invaluable for handling oscillating functions and functions that are difficult to analyze directly.