What Is the Squeeze Theorem in Calculus? Formula & Examples 2025

The squeeze theorem is a limit tool in calculus: if a function is trapped between two other functions that share the same limit at a point, then the trapped function must have that same limit. It is used when a function is hard to evaluate directly, but easy to bound. The theorem shows up under several names, including the sandwich theorem and the pinching theorem. It works for limits at a point, at infinity, and for sequences.

The Formal Statement

Suppose three functions satisfy g(x) ≤ f(x) ≤ h(x) for every x near a point a, except possibly at a itself. If both bounding functions approach the same value L as x → a, then f(x) also approaches L. The bounds g and h are called the lower and upper bounds of f.

Two features make the theorem flexible. First, the point a does not need to sit inside the interval of interest; if a is an endpoint, the limits become one-sided limits. Second, the same idea extends to infinite intervals, so statements like x → ∞ work just as well. If g(x) ≤ f(x) ≤ h(x) eventually and the outer limits are equal, the conclusion still holds.

The result also applies to sequences. If (aₙ) ≤ (bₙ) ≤ (cₙ) for all n beyond some index N, and (aₙ) and (cₙ) both converge to the same number ℓ, then (bₙ) converges to ℓ too. Because the limit of a sequence and the limit of a function are closely linked concepts, the two versions of the theorem share the same underlying logic — a point noted on the Wikipedia page for the squeeze theorem.

Why It Works

The reasoning is short enough to state plainly. If f is trapped between two functions that both close in on L, then f has nowhere else to go. More carefully, the lower limit of f must be at least L, and the upper limit of f must be at most L; since the lower limit can never exceed the upper limit, the two are forced to be equal. That common value is the limit of f.

An ε–δ proof makes the same argument with inequalities: once g and h are each within ε of L, the chain g(x) ≤ f(x) ≤ h(x) keeps f(x) within ε of L as well.

How to Use It

Applying the theorem follows a consistent three-step pattern.

1. Bound the function. Find a simpler lower function g and upper function h that sandwich the original function, often using known inequalities such as −1 ≤ sin(x) ≤ 1.
2. Take both limits. Compute lim g(x) and lim h(x) as x approaches the target point.
3. Compare. If the two limits match, that shared value is the limit of the squeezed function. If they differ, the theorem tells you nothing — a different method is needed.

The theorem is a confirmation tool, not a computation shortcut: it does not evaluate the limit directly, it verifies a value you suspect from the bounds. Lecture notes like those at Paul’s Online Math Notes cover the same technique in the context of standard limit problems.

Worked Example

Find the limit of f(x) = x² sin(1/x) as x → 0.

Start with the bound on sine: for any input, −1 ≤ sin(1/x) ≤ 1. Multiplying through by x² (which is non-negative for x ≠ 0) gives:

−x² ≤ x² sin(1/x) ≤ x²

Now take limits of the outer functions. As x → 0, both −x² and x² approach 0. The two bounds meet at the same value, so the squeeze theorem applies, and:

lim (x → 0) x² sin(1/x) = 0

That matches what the graph suggests: the oscillation of sin(1/x) never settles, but the growing x² factor flattens it toward the origin. This kind of example is standard in first-year calculus courses such as MIT OpenCourseWare, where the theorem is often introduced alongside continuity and limit laws.

The technique is not a one-trick result. It appears in several recurring places. Showing that lim (x → 0) sin(x)/x = 1 typically uses a geometric squeeze with the sine, the angle, and the tangent. Expressions like xⁿ sin(1/x) or xⁿ cos(1/x) are bounded by ±xⁿ, so the outer limits decide the outcome. Bounding a messy sequence between two simpler ones is a common way to show convergence. And bounding by functions like 1/x lets you conclude a limit of 0 as x grows without bound.

One practical caution: the bounds must hold on an interval that approaches the target point, even if they fail at the point itself. And if the two outer limits disagree, the inequality has not pinned anything down — the function may still converge, but the theorem cannot prove it.

Key Takeaways

• The squeeze theorem states that if g(x) ≤ f(x) ≤ h(x) near a point and both g and h approach the same value L, then f also approaches L.
• It works for limits at a point, one-sided limits at endpoints, limits at infinity, and convergent sequences.
• Applying it means bounding the function, taking the limits of both bounds, and checking that they match.
• It confirms a limit rather than computing one and is especially useful for oscillating or hard-to-evaluate functions.
• If the two bounding limits differ, the theorem gives no conclusion and another method is needed.