The Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) near x = a, and lim g(x) = lim h(x) = L, then lim f(x) = L.

Classic application: lim(x→0) x²sin(1/x)

-1 ≤ sin(1/x) ≤ 1

-x² ≤ x²sin(1/x) ≤ x²

lim(-x²) = 0 and lim(x²) = 0

By Squeeze Theorem, lim x²sin(1/x) = 0.

When to use: When direct evaluation fails and the function oscillates but is bounded by functions whose limits you know.

AP exam: Rare but appears occasionally. If you see sin(1/x) or cos(1/x) multiplied by something going to zero, think Squeeze Theorem.

Reference:

Wikipedia: Squeeze Theorem

https://en.wikipedia.org/wiki/Squeeze_theorem