Squeeze Theorem

College/University

Definition

Also known as the Pinching or Sandwich Theorem. It tells you that if a function that is between two functions that approach the same limit, it too must also approach that limit. In other words, a function is "sandwich"-ed between two other ones.

Worked examples

\(\frac{-x^2}{2} \le x^2 \sin\left(\frac{1}{x}\right) \le \frac{x^2}{2}\) for \(x \ne 0\). As \(x \to 0\), both bounds approach \(0\), so \(\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0\).

The middle function is squeezed between two functions that both go to zero, so it must also go to zero.

\(1 \le \frac{\sin x}{x} \cdot \cos x \le \cos x\) near \(x = 0\). Both bounds approach \(1\), so \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).

Sandwiching between bounds that meet at the same limit forces the middle function to that limit.

Common mistakes

\(g(x) \le f(x) \le h(x)\) and \(\lim g = 2\), \(\lim h = 3\), so \(\lim f\) is between \(2\) and \(3\) → The Squeeze Theorem only works when both bounds approach the *same* limit. If the bounds go to different values, the theorem does not apply and you cannot conclude anything about the middle limit.
Applying the theorem without verifying the inequality \(g(x) \le f(x) \le h(x)\) holds near the point → First prove the inequality in a neighborhood of the limit point, then apply the theorem. The sandwich must be valid near (but not necessarily at) the point where you are taking the limit.
Using the Squeeze Theorem when you can compute \(\lim f(x)\) directly → Use the theorem only when direct substitution or algebra fails. The Squeeze Theorem is a tool for tricky limits where the function oscillates or is otherwise hard to evaluate directly.

Where you'll use it next

You'll use the Squeeze Theorem to prove key trigonometric limits in Calculus I, evaluate limits of oscillating functions, and later in analysis to establish convergence of sequences and series.

Found in 1 StudyPug lesson

Mastering the Squeeze Theorem in Calculus

Calculus 1

Unlock the power of the Squeeze Theorem to solve challenging limit problems. Learn when and how to apply this essential calculus tool, with step-by-step guidance and real-world applications.

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