Squeeze Theorem — Mathematics Knowledge System

Overview

The Squeeze Theorem (also called the Sandwich Theorem or Pinching Theorem) states: if g(x) <= f(x) <= h(x) near a, and lim_{x->a} g(x) = lim_{x->a} h(x) = L, then lim_{x->a} f(x) = L. The function f is 'squeezed' between g and h, forcing its limit to equal their common limit. The Squeeze Theorem is essential for computing limits like lim_{x->0} sin(x)/x = 1 and lim_{x->0} x^2 * sin(1/x) = 0, where direct substitution fails.

Intuition

If a runner is always between two other runners who both finish at exactly 3:00, then the squeezed runner must also finish at 3:00 (or there's a contradiction). Similarly, if f is squeezed between g and h, and both g and h approach L, then f has no choice but to approach L too -- it can't escape.

Formal Definition

Definition

Suppose that g(x) <= f(x) <= h(x) for all x near a (except possibly at a itself). If lim_{x->a} g(x) = L and lim_{x->a} h(x) = L, then lim_{x->a} f(x) = L. Proof: given epsilon > 0, choose delta_1 so |g(x)-L| < epsilon when 0<|x-a|<delta_1, and delta_2 so |h(x)-L| < epsilon when 0<|x-a|<delta_2. Let delta = min(delta_1, delta_2, r) where r is the radius in which g<=f<=h holds. Then -epsilon < g(x)-L <= f(x)-L <= h(x)-L < epsilon, so |f(x)-L| < epsilon.

g(x) \le f(x) \le h(x) \text{ near } a,\quad \lim_{x\to a} g(x) = \lim_{x\to a} h(x) = L \implies \lim_{x\to a} f(x) = L

Squeeze Theorem

\lim_{x \to 0} \frac{\sin x}{x} = 1

Classic application

\lim_{x \to 0} x^2 \sin\frac{1}{x} = 0

Another application

Notation

Notation	Meaning
$g(x) \le f(x) \le h(x)$	f is squeezed between g and h
$\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L$	Common limit of the bounds

Theorems

Theorem 1: Squeeze Theorem

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

Theorem 2: Standard Trigonometric Limit

lim_{x->0} sin(x)/x = 1. Proof: for 0 < x < pi/2, using the geometric fact that sin(x) < x < tan(x) (from comparing areas of circular sector and triangles): cos(x) < sin(x)/x < 1. Both cos(x) -> 1 and 1 -> 1 as x -> 0+. By the Squeeze Theorem, sin(x)/x -> 1. Since sin(x)/x is even, the limit from the left also equals 1.

Theorem 3: Zero Times Bounded = Zero

If lim_{x->a} f(x) = 0 and g is bounded near a (|g(x)| <= M for some M and all x near a), then lim_{x->a} f(x)*g(x) = 0. Proof: squeeze -M*|f(x)| <= f(x)*g(x) <= M*|f(x)|. Both bounds go to 0.

Worked Examples

1
sin(1/x) is bounded: -1 <= sin(1/x) <= 1 for all x != 0.
2
Multiply by x^2 >= 0: -x^2 <= x^2*sin(1/x) <= x^2.
$-x^2 \le x^2 \sin(1/x) \le x^2$
3
Both bounds go to 0 as x->0: lim_{x->0} (-x^2) = 0 and lim_{x->0} x^2 = 0.
$\lim_{x\to 0}(-x^2) = \lim_{x\to 0}x^2 = 0$
4
By the Squeeze Theorem: lim_{x->0} x^2*sin(1/x) = 0.
$\lim_{x\to 0}x^2\sin\frac{1}{x} = 0$

✓ Answer

lim_{x->0} x^2*sin(1/x) = 0. Note that sin(1/x) oscillates wildly near 0, but x^2 forces the product to 0.

Practice Problems

Easyapplication

Evaluate lim_{x->0} x*cos(1/x) using the Squeeze Theorem.

Common Mistakes

Common Mistake

Thinking the Squeeze Theorem requires g and h to have the same sign as f.

The theorem only requires g <= f <= h near a, not g >= 0 or h <= 0. For example, to squeeze x^2*sin(1/x): since -x^2 <= x^2*sin(1/x) <= x^2, both bounds (including the negative one -x^2) go to 0. The bounds can be of any sign as long as they squeeze f from both sides.

Quiz

The Squeeze Theorem requires:

f, g, h to all be continuousg(x) <= f(x) <= h(x) near a, AND lim g = lim h = Lf to be differentiable at ag and h to be polynomials

Historical Background

The Squeeze Theorem was known implicitly to ancient Greek mathematicians who used the method of exhaustion (bounding areas between inscribed and circumscribed polygons). Archimedes used a version to compute the area of a circle. The formal statement in the context of limits was given by Cauchy in the 19th century. It is now a standard tool in calculus and real analysis.

250 BC
Archimedes uses a squeeze argument (method of exhaustion) to compute pi
Archimedes
1821
Cauchy formalizes squeeze-type arguments in limit theory
Augustin-Louis Cauchy

Summary

If g(x) <= f(x) <= h(x) near a and lim g = lim h = L, then lim f = L (Squeeze Theorem).
Classic use: lim_{x->0} sin(x)/x = 1 (squeezed by cos(x) < sin(x)/x < 1).
Bounded times zero: if f->0 and |g|<=M, then f*g->0 (squeeze by -M|f| <= fg <= M|f|).
Useful when direct evaluation or L'Hopital is complicated by oscillations.

References

BookStewart, J. Calculus: Early Transcendentals. 8th ed. Cengage, 2015.
BookSpivak, M. Calculus. 4th ed. Publish or Perish, 2008.
WebsiteWikipedia -- Squeeze theorem