Squeeze Theorem

The Squeeze Theorem (also called the Sandwich Theorem or Pinching Theorem) is a powerful technique in calculus for evaluating limits of functions that are difficult or impossible to compute directly. The core idea is beautifully simple: if a function f(x) is always trapped between two other functions g(x) and h(x) near a point, and both of those bounding functions approach the same limit L at that point, then f(x) must also approach L. The function is 'squeezed' into having no choice but to converge to L. This theorem is especially valuable when direct substitution, factoring, or algebraic manipulation fail to resolve a limit. The most famous application is proving that lim(x→0) sin(x)/x = 1, a result that underpins the derivatives of all trigonometric functions. Without the Squeeze Theorem, this foundational limit would be extremely difficult to establish rigorously. Understanding the Squeeze Theorem deepens your appreciation for how mathematicians establish results with certainty. Rather than just calculating, you construct an argument: build two simple functions that bound the tricky one, show they share a limit, and conclude the trapped function must match. This style of reasoning — using known, controlled quantities to pin down unknown ones — appears throughout higher mathematics and mathematical physics.

Key takeaways

State the Setup
Verify the Bounding Limits
Apply the Theorem
Choose Bounds Strategically
One-Sided and Infinite Limits

State the Setup

You need three functions: a lower bound g(x), the target function f(x), and an upper bound h(x). The inequality g(x) ≤ f(x) ≤ h(x) must hold for all x near (but not necessarily at) the point a you are approaching. This means there exists some open interval around a where the inequality is satisfied — x = a itself can be excluded.

Verify the Bounding Limits

Compute the limits of both bounding functions as x → a. If lim(x→a) g(x) = L and lim(x→a) h(x) = L (the same value L for both), you satisfy the second condition of the theorem. These limits are usually straightforward to evaluate by direct substitution because g and h are chosen to be simple functions.

Apply the Theorem

The Squeeze Theorem guarantees that lim(x→a) f(x) = L. You do not need to find a formula for f(x)'s limit independently — the squeezing argument does the work. Write a clear conclusion: 'Since g(x) ≤ f(x) ≤ h(x) near x = a, and both bounding limits equal L, by the Squeeze Theorem lim(x→a) f(x) = L.'

Choose Bounds Strategically

The art of the Squeeze Theorem lies in choosing useful bounds. A common strategy exploits the fact that -1 ≤ sin(θ) ≤ 1 and -1 ≤ cos(θ) ≤ 1 for all θ. If f(x) contains a bounded oscillating factor like sin(1/x) multiplied by something that goes to 0, you can replace the oscillating part with ±1 to create your bounding functions. Always verify the inequality holds rigorously near the point of interest.

One-Sided and Infinite Limits

The Squeeze Theorem also works for one-sided limits (x → a⁺ or x → a⁻) and for limits at infinity (x → ∞ or x → -∞). As long as the bounding inequality holds in the appropriate region and both bounding functions converge to the same value L, the conclusion applies. For example, if g(x) ≤ f(x) ≤ h(x) for all x > M for some large M, and both bounds approach L as x → ∞, then f(x) → L as well.

The Key Equation

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

f(x) — Target function (dimensionless): The function whose limit we want to find but cannot compute directly.
g(x) — Lower bound function (dimensionless): A simpler function satisfying g(x) ≤ f(x) near x = a, whose limit can be evaluated easily.
h(x) — Upper bound function (dimensionless): A simpler function satisfying f(x) ≤ h(x) near x = a, whose limit can be evaluated easily.
L — Common limit (dimensionless): The shared limit value that both bounding functions approach, which f(x) is forced to match.
a — Point of approach (dimensionless): The x-value (or ∞) that x is approaching. The function f need not be defined at x = a.

Worked Example

Problem. Evaluate lim(x→0) x² sin(1/x).

\lim_{x \to 0} x^2 \sin\!\left(\frac{1}{x}\right) = 0

Answer. The limit equals 0.

Figure

Diagram

Origins · 1 minAugustin-Louis Cauchy and how this was discovered

Augustin-Louis Cauchy

1789–1857 · French

Cauchy was one of the founders of rigorous mathematical analysis. He formalized the definition of a limit in the early 19th century, replacing intuitive notions of 'infinitely small quantities' with precise epsilon-delta arguments. The Squeeze Theorem and related comparison results for limits were developed within this framework that Cauchy helped establish. He authored the influential textbook Cours d'analyse (1821), which set the standard for rigorous calculus for generations. Cauchy was extraordinarily prolific, publishing around 800 research articles — so many that the French Academy of Sciences eventually imposed a page limit on submissions specifically because of him. His name appears on more theorems and mathematical concepts than almost any other mathematician in history.

Definition of a Limit — The Squeeze Theorem is built on the formal definition of a limit; understanding epsilon-delta definitions makes the theorem's proof fully rigorous.
Limit Laws — Limit laws (sum, product, constant multiple rules) are used to compute the bounding limits in the Squeeze Theorem argument.
Special Trigonometric Limits — The classic result lim(x→0) sin(x)/x = 1 is the most important application of the Squeeze Theorem and forms the basis for differentiating trigonometric functions.
Continuity of Functions — Squeeze Theorem results help verify continuity at tricky points, such as showing x²sin(1/x) extended by f(0)=0 is continuous at the origin.
Derivatives of Trigonometric Functions — The derivative of sin(x) requires the limit lim(x→0) sin(x)/x, which is proven using the Squeeze Theorem, making this theorem essential groundwork for trigonometric differentiation.