Squeeze Theorem Formula

The Formula

If g(x) \leq f(x) \leq h(x) and \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, then \lim_{x \to a} f(x) = L.

When to use: If f is squeezed between two functions that both approach the same value L, then f has no choice—it must also approach L. Like being caught between two walls closing in to the same point.

Quick Example

Find \lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right).
Since -1 \leq \sin\left(\frac{1}{x}\right) \leq 1, we have -x^2 \leq x^2\sin\left(\frac{1}{x}\right) \leq x^2.
Both -x^2 \to 0 and x^2 \to 0, so \lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0

Notation

g(x) \leq f(x) \leq h(x) — g is the lower bound, h is the upper bound, and f is squeezed between them.

What This Formula Means

If g(x) \leq f(x) \leq h(x) near x = a, and \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, then \lim_{x \to a} f(x) = L.

If f is squeezed between two functions that both approach the same value L, then f has no choice—it must also approach L. Like being caught between two walls closing in to the same point.

Formal View

If \exists \delta_0 > 0 such that g(x) \leq f(x) \leq h(x) for all x with 0 < |x - a| < \delta_0, and \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, then \lim_{x \to a} f(x) = L.

Worked Examples

Example 1

easy
Use the squeeze theorem to find \displaystyle\lim_{x \to 0} x^2 \sin\!\left(\frac{1}{x}\right).

Solution

  1. 1
    Since -1 \leq \sin\!\left(\frac{1}{x}\right) \leq 1 for all x \neq 0:
  2. 2
    Multiply by x^2 \geq 0: -x^2 \leq x^2\sin\!\left(\frac{1}{x}\right) \leq x^2.
  3. 3
    \lim_{x\to 0}(-x^2) = 0 and \lim_{x\to 0}(x^2) = 0.
  4. 4
    By the squeeze theorem: \lim_{x\to 0} x^2\sin\!\left(\frac{1}{x}\right) = 0.

Answer

0
The function \sin(1/x) oscillates wildly near 0, so its limit doesn't exist alone. Multiplying by x^2 traps it between -x^2 and x^2, both going to 0, so the product must also go to 0.

Example 2

hard
Use the squeeze theorem to prove \displaystyle\lim_{x \to 0} \frac{\sin x}{x} = 1.

Common Mistakes

  • Trying to use the Squeeze Theorem when the upper and lower bounds don't have the same limit—the theorem only works when both bounds approach the same value L.
  • Forgetting that the inequality g(x) \leq f(x) \leq h(x) only needs to hold near x = a, not everywhere. The behavior far from a is irrelevant.
  • Attempting to evaluate \lim_{x \to 0} \sin(1/x) directly instead of recognizing it oscillates—the Squeeze Theorem is needed when \sin(1/x) is multiplied by something that goes to zero.

Why This Formula Matters

Used to prove the fundamental limit \lim_{x \to 0} \frac{\sin x}{x} = 1, which underlies all of trigonometric calculus. Also essential for limits involving oscillating functions multiplied by vanishing terms.

Frequently Asked Questions

What is the Squeeze Theorem formula?

If g(x) \leq f(x) \leq h(x) near x = a, and \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, then \lim_{x \to a} f(x) = L.

How do you use the Squeeze Theorem formula?

If f is squeezed between two functions that both approach the same value L, then f has no choice—it must also approach L. Like being caught between two walls closing in to the same point.

What do the symbols mean in the Squeeze Theorem formula?

g(x) \leq f(x) \leq h(x) — g is the lower bound, h is the upper bound, and f is squeezed between them.

Why is the Squeeze Theorem formula important in Math?

Used to prove the fundamental limit \lim_{x \to 0} \frac{\sin x}{x} = 1, which underlies all of trigonometric calculus. Also essential for limits involving oscillating functions multiplied by vanishing terms.

What do students get wrong about Squeeze Theorem?

The hard part is finding the bounding functions g and h. For oscillating functions like \sin or \cos, use -1 \leq \sin(\cdot) \leq 1 and multiply by the vanishing factor.

What should I learn before the Squeeze Theorem formula?

Before studying the Squeeze Theorem formula, you should understand: limit.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Limits Explained Intuitively: The Foundation of Calculus →
← Back to Squeeze Theorem See All Examples Practice Problems