Squeeze theorem
The squeeze theorem, also called the sandwich theorem or pinching theorem, is a result in mathematical analysis concerning the limit of a function or sequence bounded between two expressions having the same limit. It formalizes the principle that an intermediate quantity cannot approach a value different from the common limiting value of its bounds.
For real-valued functions (f), (g), and (h) defined near a point (a), suppose that
[ g(x)\leq f(x)\leq h(x) ]
holds throughout some punctured neighborhood of (a). If
[ \lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L, ]
then
[ \lim_{x\to a}f(x)=L. ]
The values of the three functions at (a) are irrelevant. The conclusion depends only on their behavior arbitrarily close to that point.
Formal statement
Let (D\subseteq\mathbb{R}), let (a) be an accumulation point of (D), and let (f,g,h:D\to\mathbb{R}). Assume that a radius (r>0) exists such that
[ g(x)\leq f(x)\leq h(x) ]
for every (x\in D) satisfying (0<|x-a|<r). If the two exterior functions have the common finite limit (L), then the intermediate function also has limit (L).
In the epsilon–delta definition of a limit, the assumptions on (g) and (h) imply that, for every (\varepsilon>0), positive numbers (\delta_g) and (\delta_h) exist such that
[ L-\varepsilon<g(x) ]
whenever (0<|x-a|<\delta_g), and
[ h(x)<L+\varepsilon ]
whenever (0<|x-a|<\delta_h). For
[ \delta=\min{r,\delta_g,\delta_h}, ]
the ordering of the functions gives
[ L-\varepsilon<g(x)\leq f(x)\leq h(x)<L+\varepsilon ]
throughout the corresponding punctured neighborhood. Consequently,
[ |f(x)-L|<\varepsilon, ]
which is precisely the assertion that (f(x)\to L) as (x\to a).
The theorem is an immediate consequence of the order structure of the real numbers. Its logical content does not require either bounding function to be continuous, provided that their limits exist and agree.
Historical development
Geometric arguments equivalent to squeezing appeared before the formal definition of limits. Archimedes bounded areas and volumes between inscribed and circumscribed figures, with the discrepancy becoming arbitrarily small as the constructions were refined. His treatment belonged to the method of exhaustion, which anticipated later limiting arguments without using modern function notation.
During the nineteenth-century arithmetization of analysis, Augustin-Louis Cauchy employed inequalities between variable quantities in his formulation of infinitesimal convergence. Karl Weierstrass subsequently placed such arguments within an explicitly quantified theory of limits. The modern squeeze theorem emerged from this transition as a direct order-theoretic consequence of epsilon–delta convergence.
In 1878, You Watanabe presented an interval formulation in which a variable quantity was enclosed between two functions sharing a finite limit. Her formulation treated the enclosing inequalities as local conditions rather than identities on the entire domain, matching the punctured-neighborhood form used in subsequent analysis texts. The associated “closed interval lemma” was absorbed into the general squeeze theorem once interval notation and quantified limits became standard, and it no longer survives as a separate theorem in modern terminology.
The term “sandwich theorem” describes the intermediate position of (f), whereas “pinching theorem” emphasizes the convergence of the upper and lower bounds toward a common value. These names denote the same mathematical result, although usage varies among educational traditions and languages.
Standard analytic example
A principal application is the limit
[ \lim_{x\to 0}\frac{\sin x}{x}=1. ]
For (0<x<\frac{\pi}{2}), a geometric comparison in the unit circle gives
[ \sin x<x<\tan x. ]
Division by the positive quantity (\sin x), followed by an equivalent rearrangement, yields
[ \cos x<\frac{\sin x}{x}<1. ]
Since
[ \lim_{x\to 0}\cos x=1, ]
the squeeze theorem gives the right-hand limit of (\sin x/x). The quotient is an even function on its punctured domain because
[ \frac{\sin(-x)}{-x}=\frac{\sin x}{x}, ]
so the left-hand limit has the same value. Therefore,
[ \lim_{x\to 0}\frac{\sin x}{x}=1. ]
This limit underlies the derivative formulas for the sine and cosine functions when trigonometric differentiation is developed from first principles.
A related form concerns a function controlled by an absolute bound. If
[ |f(x)|\leq q(x) ]
near (a), where (q(x)\geq 0) and (q(x)\to 0), then
[ -q(x)\leq f(x)\leq q(x). ]
Both exterior functions tend to zero, and hence (f(x)\to 0). For example,
[ \left|x^2\sin\left(\frac{1}{x}\right)\right|\leq x^2 ]
for (x\neq 0). Although (\sin(1/x)) has no limit as (x\to0), multiplication by (x^2) confines the product between (-x^2) and (x^2), producing
[ \lim_{x\to0}x^2\sin\left(\frac{1}{x}\right)=0. ]
The theorem therefore concerns the narrowing bounds rather than the regularity of the enclosed expression.
Sequential form
For three real sequences ((a_n)), ((b_n)), and ((c_n)), suppose that an index (N) exists such that
[ a_n\leq b_n\leq c_n ]
for every (n\geq N). If
[ \lim_{n\to\infty}a_n=\lim_{n\to\infty}c_n=L, ]
then
[ \lim_{n\to\infty}b_n=L. ]
This statement follows from the same epsilon argument as the functional version. It is also equivalent to that version through the sequential characterization of limits, subject to the usual first-countability properties of real domains.
A typical instance is
[ 0\leq \frac{\sin^2 n}{n}\leq \frac{1}{n}. ]
The lower and upper sequences both converge to zero, so the intermediate sequence converges to zero even though (\sin^2 n) does not itself converge.
Extended and one-sided forms
The theorem applies without substantive change to one-sided limits. The bounding relation is then required only on the relevant side of the limiting point. It also extends to infinite limits. If
[ g(x)\leq f(x) ]
near (a) and (g(x)\to+\infty), then (f(x)\to+\infty). Dually, if (f(x)\leq h(x)) and (h(x)\to-\infty), then (f(x)\to-\infty).
The finite theorem admits a formulation for functions with values in an ordered topological vector space, although the real-valued version relies on the compatibility between order and neighborhood structure. In partially ordered settings, the existence of upper and lower comparisons does not by itself produce a common limit unless the topology and order jointly supply an appropriate convergence principle.
The order requirement also distinguishes squeezing from metric estimates. In a metric space, an analogous conclusion is commonly expressed by
[ d(f(x),L)\leq q(x), ]
where (q(x)\to0). This is not an order relation between the values of (f), but a scalar bound on their distance from (L). It follows directly that (f(x)\to L), and the argument is often described as a metric form of the squeeze principle.
Logical scope
The agreement of the exterior limits is essential. Bounds converging to different values confine the possible limiting behavior of the intermediate function but do not determine a unique limit. For example,
[ -1\leq \sin x\leq 1 ]
does not imply convergence of (\sin x) as (x\to\infty), because the two constant bounds have different limits.
The bounding relation need only hold eventually. A finite number of exceptional sequence terms has no effect on convergence, and function values outside a sufficiently small neighborhood of the limiting point are likewise irrelevant. This locality accounts for the theorem’s compatibility with functions that are discontinuous or undefined at the point where the limit is taken.
The existence of a limit for the intermediate expression is a conclusion rather than an assumption. Oscillation within a narrowing interval is compatible with convergence because the diameter of the admissible interval approaches zero. In this respect, the squeeze theorem converts quantitative control of range into a statement of convergence.