1.8 The Squeeze Theorem

The idea

The Squeeze Theorem (Sandwich Theorem): if g(x) ≤ f(x) ≤ h(x) near c, and lim g = lim h = L, then f is trapped and lim f = L too.

Classic uses

It proves limits that resist algebra, like lim(x→0) sin(x)/x = 1 and lim(x→0) x²·sin(1/x) = 0 (since −x² ≤ x²sin(1/x) ≤ x², and both bounds → 0).

Key terms

Squeeze Theorem

If g ≤ f ≤ h near c and lim g = lim h = L, then lim f = L.

Also called

The Sandwich Theorem.

Key condition

The two bounding functions must have the same limit at c.

sin(x)/x

lim(x→0) sin(x)/x = 1.

x²·sin(1/x)

lim(x→0) x²·sin(1/x) = 0 by squeezing between ±x².

Try an AP-style question

If 1 − x² ≤ f(x) ≤ 1 + x² for all x near 0, what is lim(x→0) f(x)?

1. 0
2. Cannot be determined
3. 1
4. 2