Squeeze Theorem

Calculus
principle

Also known as: sandwich theorem, pinching theorem

Grade 9-12

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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. Used to prove the fundamental limit \lim_{x \to 0} \frac{\sin x}{x} = 1, which underlies all of trigonometric calculus.

This concept is covered in depth in our evaluating limits step by step, with worked examples, practice problems, and common mistakes.

Definition

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.

💡 Intuition

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.

🎯 Core Idea

The Squeeze Theorem is a tool for finding limits that can't be computed by direct substitution or algebraic manipulation. Trap the unknown function between two known ones.

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

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.

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.

🌟 Why It 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.

💭 Hint When Stuck

Bound the oscillating part between -1 and 1, then multiply the entire inequality by the non-oscillating factor.

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.

🚧 Common Stuck Point

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.

⚠️ 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.

Frequently Asked Questions

What is Squeeze Theorem in Math?

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.

What is the Squeeze Theorem 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 do you use Squeeze Theorem?

Bound the oscillating part between -1 and 1, then multiply the entire inequality by the non-oscillating factor.

Prerequisites

limit

Next Steps

continuity types

How Squeeze Theorem Connects to Other Ideas

To understand squeeze theorem, you should first be comfortable with limit. Once you have a solid grasp of squeeze theorem, you can move on to continuity types.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Limits Explained Intuitively: The Foundation of Calculus →
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