Squeeze Theorem

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Squeeze Theorem: meaning, method and solved example (AP Calculus AB) – Maths PDF

In short: The squeeze theorem says that if g(x) ≤ f(x) ≤ h(x) near c and g and h both approach L, then f is forced to approach L as well.

AP Calculus ABCalculusAP Calculus AB Unit 1

What is Squeeze Theorem?

The squeeze theorem works like a sandwich. If a function is trapped between a lower bound and an upper bound, and both bounds head to the same value, the function caught in the middle has nowhere else to go.

Formally, suppose \(g(x) \le f(x) \le h(x)\) for all \(x\) near \(c\), except possibly at \(c\). If \(\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L\), then \(\lim_{x \to c} f(x) = L\). It is the tool of choice for wild oscillating functions, where direct substitution fails.

How to spot it

  • A function contains sin or cos of an expression like 1/x, which oscillates wildly near the target point, a classic case for the squeeze theorem.
  • You know the bounds -1 ≤ sin(u) ≤ 1 and can multiply them by a simpler factor.
  • The problem gives two inequalities around f and asks for the value f must approach.
  • A graph shows a wiggly curve sandwiched between two smooth curves that meet at a point.

The method

  1. Find bounds for the troublesome part, for example \(-1 \le \sin\frac{1}{x} \le 1\).
  2. Multiply by the other factor, taking care to flip inequalities if it can be negative, to get \(g(x) \le f(x) \le h(x)\).
  3. Compute the values of \(g\) and \(h\) as \(x\) approaches \(c\) and confirm they agree.
  4. Conclude, by the squeeze theorem, that \(f\) approaches the same number.
upper: y = x²lower: y = -x²both reach 0f(x) = x² sin(1/x)
The wiggly curve is trapped between y = x² and y = -x², so it must reach 0.

Squeeze Theorem: solved example

Problem

Use the theorem to find \(\lim_{x \to 0} x^2\sin\dfrac{1}{x}\).

Solution

Sine of anything lies between -1 and 1, so \(-1 \le \sin\frac{1}{x} \le 1\). Multiplying by \(x^2 \ge 0\) keeps the order: \(-x^2 \le x^2\sin\frac{1}{x} \le x^2\).

Both \(-x^2\) and \(x^2\) approach 0 as \(x \to 0\). The middle function is squeezed, so the value is 0.

Try it

Question 1

Find \(\lim_{x \to 0} x\sin\dfrac{5}{x}\).

Show the answer

From \(-1 \le \sin\frac{5}{x} \le 1\) we get \(-|x| \le x\sin\frac{5}{x} \le |x|\); both bounds tend to 0, so the value is 0.

Question 2

Near \(x = 1\), \(2x \le g(x) \le x^2 + 1\). What is \(\lim_{x \to 1} g(x)\)?

Show the answer

At \(x = 1\) the lower bound is 2 and the upper bound is \(1 + 1 = 2\). So the value is 2.

Squeeze Theorem vs. Limit

A limit is the general idea of approaching a value. The sandwich result is one particular technique for computing it, used when the function itself is too jumpy for direct substitution or algebra. It also requires both outer functions to approach the very same number, not merely nearby numbers.

Limit

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