The Squeeze theorem, also known as the sandwich theorem or pinching theorem, is a fundamental theorem in mathematical analysis and calculus that establishes the limit of a function by demonstrating that it is bounded between two other functions approaching the same value.[1] Formally, the theorem states that if g(x) \leq f(x) \leq h(x) for all x in some open interval containing c except possibly at x = c, and if \lim_{x \to c} g(x) = L and \lim_{x \to c} h(x) = L, then \lim_{x \to c} f(x) = L.[1] This result is particularly valuable for evaluating limits that resist direct algebraic computation, such as oscillatory or indeterminate forms.[2]One of the theorem's most notable applications arises in the evaluation of trigonometric limits essential to differential calculus, including \lim_{x \to 0} \frac{\sin x}{x} = 1, which is obtained by squeezing the function between geometric bounds derived from the unit circle.[3] By establishing this limit, the Squeeze theorem facilitates the derivation of the derivatives of sine and cosine, confirming that \frac{d}{dx} \sin x = \cos x and \frac{d}{dx} \cos x = -\sin x at x = 0.[3] Beyond basic limits, the theorem extends to multivariable calculus and real analysis, where it aids in proving continuity and properties of bounded sequences.[4][5] For instance, it can bound expressions like x^2 \cos(1/x) between -x^2 and x^2 as x \to 0, yielding a limit of 0 despite the rapid oscillations of \cos(1/x).[1]
Formal Statement
For Real-Valued Functions
The squeeze theorem for real-valued functions states that if a function f is bounded between two other functions g and h in a deleted neighborhood of a point a, and if the limits of g and h as x approaches a both equal the same value L, then the limit of f as x approaches a also equals L.[6][7]More precisely, let f, g, and h be real-valued functions defined on some open interval containing the real number a, except possibly at a itself. Suppose there exists \delta > 0 such that g(x) \leq f(x) \leq h(x) for all x \in (a - \delta, a + \delta) \setminus \{a\}, and suppose\lim_{x \to a} g(x) = L, \quad \lim_{x \to a} h(x) = L.Then\lim_{x \to a} f(x) = L.[6][7][8]This formulation addresses two-sided limits, which serve as the default case; analogous statements hold for one-sided limits and are treated in a separate section.[6][8]
For Sequences
The squeeze theorem for sequences, also known as the sandwich theorem in some mathematical texts, provides a method to determine the limit of a sequence by bounding it between two other sequences that converge to the same value.[9][10] Specifically, suppose there exist sequences \{g_n\}, \{f_n\}, and \{h_n\} such that g_n \leq f_n \leq h_n for all n \geq N, where N is some positive integer, and \lim_{n \to \infty} g_n = \lim_{n \to \infty} h_n = L. Then, \lim_{n \to \infty} f_n = L.[9][11] This formulation emphasizes convergence in discrete terms, where the inequality need only hold for sufficiently large indices n, allowing the initial terms (for n < N) to behave arbitrarily without affecting the limit.[9][10]The condition of "sufficiently large n" is formalized by the existence of such an N \in \mathbb{N}, ensuring the bounding sequences "squeeze" the target sequence toward the common limit L as n increases indefinitely.[11][12] This discrete version aligns closely with the squeeze theorem for real-valued functions but treats the domain as the natural numbers with the index n \to \infty, effectively viewing sequences as a special case of functions restricted to integer inputs.[9] The theorem is particularly useful for establishing convergence when direct computation of \lim_{n \to \infty} f_n is challenging, relying instead on the known limits of the bounding sequences.[13]
Proof
Epsilon-Delta Proof for Functions
The epsilon-delta proof of the squeeze theorem for real-valued functions proceeds as follows. Suppose g(x) \leq f(x) \leq h(x) for all x in some open interval containing a, except possibly at x = a, and that \lim_{x \to a} g(x) = L = \lim_{x \to a} h(x). To show \lim_{x \to a} f(x) = L, let \epsilon > 0 be given.[14]By the definition of the limit, there exists \delta_1 > 0 such that if $0 < |x - a| < \delta_1, then |g(x) - L| < \epsilon, which implies L - \epsilon < g(x) < L + \epsilon. Similarly, there exists \delta_2 > 0 such that if $0 < |x - a| < \delta_2, then L - \epsilon < h(x) < L + \epsilon.[14]Let \delta = \min(\delta_1, \delta_2). Then, for $0 < |x - a| < \delta, both inequalities hold simultaneously, so L - \epsilon < g(x) < L + \epsilon and L - \epsilon < h(x) < L + \epsilon. Since g(x) \leq f(x) \leq h(x), it follows thatL - \epsilon < g(x) \leq f(x) \leq h(x) < L + \epsilon,which implies L - \epsilon < f(x) < L + \epsilon, or equivalently, |f(x) - L| < \epsilon.[14]Thus, by the epsilon-delta definition of the limit, \lim_{x \to a} f(x) = L. This direct bounding argument relies on the inequality sandwiching f(x) between g(x) and h(x), without requiring the triangle inequality.[14]
Proof for Sequences
To prove the squeeze theorem for sequences, suppose \lim_{n \to \infty} g_n = L and \lim_{n \to \infty} h_n = L, and that there exists some integer N_0 such that g_n \leq f_n \leq h_n for all n > N_0.[15][9]Let \epsilon > 0. By the definition of the limit, there exists an integer N_1 > 0 such that for all n > N_1, |g_n - L| < \epsilon, which implies L - \epsilon < g_n < L + \epsilon. Similarly, there exists an integer N_2 > 0 such that for all n > N_2, |h_n - L| < \epsilon, which implies L - \epsilon < h_n < L + \epsilon.[15][9]Let N = \max\{N_0, N_1, N_2\}. Then, for all n > N,L - \epsilon < g_n \leq f_n \leq h_n < L + \epsilon.This implies L - \epsilon < f_n < L + \epsilon, or equivalently, |f_n - L| < \epsilon. Therefore, \lim_{n \to \infty} f_n = L.[15][9]To bound |f_n - L| explicitly, note that f_n \geq g_n > L - \epsilon shows f_n - L > -\epsilon, and f_n \leq h_n < L + \epsilon shows f_n - L < \epsilon. Thus, -\epsilon < f_n - L < \epsilon, confirming |f_n - L| < \epsilon.[15][9]This proof mirrors the structure of the epsilon-delta proof for functions but adapts to the discrete nature of sequences by using integer indices n and a threshold N instead of a continuous variable and delta.[15][9]
Historical Background
Ancient and Early Uses
The method of exhaustion, an early precursor to the squeeze theorem's bounding principle, was developed by the ancient Greek mathematician Eudoxus of Cnidus around 370 BC to rigorously determine areas and volumes without invoking infinitesimals or paradoxes of infinity.[16] Eudoxus applied this technique to prove key results, such as the volume of a pyramid being one-third that of a prism with the same base and height, and the volume of a cone being one-third that of a cylinder with the same base and height, by successively approximating the figures with simpler polygonal or polyhedral shapes that "exhausted" the target region from within or without.[16] These arguments relied on intuitive comparisons of magnitudes rather than formal limits, establishing bounds that narrowed toward the true value through geometric refinement.[17]Archimedes of Syracuse, building on Eudoxus' foundations in the 3rd century BC, extensively employed the method of exhaustion for more complex calculations, including areas under curves and volumes of solids of revolution.[18] A prominent example is his approximation of the area of a circle in Measurement of a Circle, where he "squeezed" the circle's area between that of an inscribed regular polygon and a circumscribed one, iteratively increasing the number of sides to bound π between $3 \frac{10}{71} and $3 \frac{1}{7}.[18] Similarly, in Quadrature of the Parabola, Archimedes bounded the area of a parabolic segment by inscribing triangles and exhausting the remainder, demonstrating the area's equality to \frac{4}{3} times the area of the inscribed triangle with the same base and height, again through successive approximations without a concept of limits.[18] These applications highlighted the power of bounding arguments in geometry, though they remained tied to axiomatic proofs in Euclid's style rather than analytic methods.[17]The intuitive bounding techniques of Eudoxus and Archimedes persisted through the medieval period via Arabic translations of Greek texts, influencing Islamic mathematicians in quadrature problems—efforts to square curved figures like circles and parabolas—where similar exhaustion-like approximations aided in area computations. This legacy carried into the Renaissance, as European scholars rediscovered Archimedes' works; for instance, Evangelista Torricelli in 1644 revisited the quadrature of the parabola using indivisibles inspired by exhaustion, providing multiple proofs that echoed ancient bounding strategies while bridging toward infinitesimal methods.[19]
Formalization in the 19th Century
The formalization of the squeeze theorem, also known as the sandwich theorem, emerged in the early 19th century as part of the broader push toward rigorous definitions of limits in mathematical analysis. Carl Friedrich Gauss contributed to the understanding of convergence in his 1813 study of hypergeometric series, where bounding arguments akin to the squeeze theorem were employed to establish convergence properties. This marked a shift from intuitive geometric methods to algebraic rigor, laying groundwork for the theorem's modern algebraic formulation.[20]Bernard Bolzano, in 1817, provided an early rigorous definition of continuity and limits using a precursor to the epsilon-delta approach, paving the way for later developments.)Augustin-Louis Cauchy further advanced this development in his seminal 1821 textbook Cours d'analyse de l'École Royale Polytechnique, where he incorporated bounding inequalities into the theory of limits. Cauchy utilized epsilon-like arguments to define limits through the algebra of inequalities, effectively embedding squeeze-like techniques to prove convergence of functions and sequences bounded between two others approaching the same value. His approach emphasized logical precision, transforming earlier informal ideas into a systematic framework for analysis.[21][22]These contributions profoundly influenced Karl Weierstrass's lectures on calculus in the 1860s, where he established the epsilon-delta definition of limits and continuity, integrating bounding methods such as the squeeze theorem into the foundations of real analysis. Weierstrass's rigorous treatment ensured the theorem's place as a core tool for proving limits without relying on infinitesimals, solidifying its role in modern mathematics.[21]
Illustrative Examples
Basic Limit of a Function
One classic application of the squeeze theorem arises when evaluating the limit \lim_{x \to 0} x^2 \sin(1/x), where direct substitution of x = 0 is indeterminate because \sin(1/x) oscillates infinitely often between -1 and 1 without approaching a single value.[23] To resolve this, note that since |\sin(1/x)| \leq 1 for all x \neq 0, it follows that -1 \leq \sin(1/x) \leq 1, and multiplying through by the nonnegative quantity x^2 yields the inequality -x^2 \leq x^2 \sin(1/x) \leq x^2.[24]The squeeze theorem for real-valued functions states that if f(x) \leq g(x) \leq h(x) for all x in some deleted neighborhood of a (except possibly at a) and \lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L, then \lim_{x \to a} g(x) = L. Applying this here, observe that \lim_{x \to 0} (-x^2) = 0 and \lim_{x \to 0} x^2 = 0, so by the squeeze theorem, \lim_{x \to 0} x^2 \sin(1/x) = 0.[23]Intuitively, the rapid oscillations of \sin(1/x) are confined within the narrowing envelope defined by -x^2 and x^2, both of which approach zero as x nears the origin; this "squeezing" forces the function values to converge to zero despite the unbounded frequency of oscillation.[24] Graphically, the plot of x^2 \sin(1/x) shows dense wiggles that diminish in amplitude toward the x-axis, illustrating how the bounding curves pinch the graph to the limit point.[25]
Trigonometric Limit
One of the most famous applications of the squeeze theorem is evaluating the limit \lim_{\theta \to 0} \frac{\sin \theta}{\theta}, which equals 1 and serves as a foundational result in calculus for derivatives of trigonometric functions.[26] This limit is established using a geometric argument on the unit circle, where the squeeze theorem bounds the expression between two functions that both approach the same value as \theta approaches 0.Consider the unit circle centered at the origin, with a radius of 1. For $0 < \theta < \frac{\pi}{2}, identify three regions: the smaller triangle formed by the origin, the point (1, 0) on the positive x-axis, and the point (\cos \theta, \sin \theta) on the circle; the circular sector bounded by the radii to (1, 0) and (\cos \theta, \sin \theta), along with the arc between them; and the larger triangle formed by the origin, (1, 0), and the point where the tangent line at (1, 0) intersects the line from the origin through (\cos \theta, \sin \theta), which is at (1, \tan \theta).[7] The areas of these regions satisfy \frac{1}{2} \sin \theta < \frac{1}{2} \theta < \frac{1}{2} \tan \theta, since the smaller triangle is contained within the sector, which is contained within the larger triangle.[26]Dividing through by \frac{1}{2} yields \sin \theta < \theta < \tan \theta. Dividing all parts by \theta > 0 gives \frac{\sin \theta}{\theta} < 1 < \frac{\tan \theta}{\theta}. Substituting \tan \theta = \frac{\sin \theta}{\cos \theta} into the right inequality produces \frac{\sin \theta}{\theta} < 1 < \frac{\sin \theta}{\theta \cos \theta}. Rearranging terms leads to \cos \theta < \frac{\sin \theta}{\theta} < 1 for $0 < \theta < \frac{\pi}{2}.[7] The same inequality \cos \theta < \frac{\sin \theta}{\theta} < 1 holds for -\frac{\pi}{2} < \theta < 0, because \frac{\sin \theta}{\theta} and \cos \theta are even functions.[26]As \theta \to 0, the bounding functions satisfy \lim_{\theta \to 0} \cos \theta = 1 and \lim_{\theta \to 0} 1 = 1. By the squeeze theorem, it follows that \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1.[7]
Sequence Convergence
The squeeze theorem extends naturally to sequences, stating that if there exist sequences b_n and c_n such that b_n \leq a_n \leq c_n for all sufficiently large n, and \lim_{n \to \infty} b_n = \lim_{n \to \infty} c_n = L, then \lim_{n \to \infty} a_n = L.[10]A classic example demonstrating this application is the sequence a_n = \frac{\sin n}{n}, where n is a positive integer. To determine its limit as n \to \infty, observe that the sine function is bounded: -1 \leq \sin n \leq 1 for all real n. Dividing by n > 0 yields the inequalities-\frac{1}{n} \leq \frac{\sin n}{n} \leq \frac{1}{n}.Since \lim_{n \to \infty} -\frac{1}{n} = 0 and \lim_{n \to \infty} \frac{1}{n} = 0, the squeeze theorem implies \lim_{n \to \infty} \frac{\sin n}{n} = 0.[10][27]This convergence holds despite the fact that \sin n itself does not converge and instead oscillates indefinitely between -1 and 1, with values dense in [-1, 1] due to the irrationality of \pi, which ensures that the sequence n \mod 2\pi is dense in [0, 2\pi). The bounding term $1/n forces the amplitude of these oscillations to diminish toward zero.[28]
Multivariable Application
The squeeze theorem applies to multivariable limits in a manner analogous to its single-variable counterpart. For functions f: \mathbb{R}^2 \to \mathbb{R}, if there exist functions g and h such that g(x,y) \leq f(x,y) \leq h(x,y) in a punctured neighborhood of (a,b) and \lim_{(x,y) \to (a,b)} g(x,y) = \lim_{(x,y) \to (a,b)} h(x,y) = L, then \lim_{(x,y) \to (a,b)} f(x,y) = L.[5] This extension relies on the multivariable limitdefinition, where the limit is L if for every \epsilon > 0, there exists \delta > 0 such that $0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta implies |f(x,y) - L| < \epsilon.[29]A representative application arises in evaluating \lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2 + y^2}}. To determine this limit using the squeeze theorem, first observe that |xy| \leq \frac{1}{2}(x^2 + y^2) by the AM-GM inequality. Thus,\left| \frac{xy}{\sqrt{x^2 + y^2}} \right| \leq \frac{\frac{1}{2}(x^2 + y^2)}{\sqrt{x^2 + y^2}} = \frac{1}{2} \sqrt{x^2 + y^2}.This yields -\frac{1}{2} \sqrt{x^2 + y^2} \leq \frac{xy}{\sqrt{x^2 + y^2}} \leq \frac{1}{2} \sqrt{x^2 + y^2}. Since \lim_{(x,y) \to (0,0)} \sqrt{x^2 + y^2} = 0, both bounding functions approach 0, so the limit is 0 by the squeeze theorem.[30] (Note: While this specific bounding uses standard inequalities, analogous applications appear in multivariable calculus texts.)[29]Switching to polar coordinates provides another perspective: let x = r \cos \theta and y = r \sin \theta, where r = \sqrt{x^2 + y^2}. Then,\frac{xy}{\sqrt{x^2 + y^2}} = r \cos \theta \sin \theta = \frac{r}{2} \sin 2\theta.Hence,\left| \frac{xy}{\sqrt{x^2 + y^2}} \right| = \frac{r}{2} |\sin 2\theta| \leq \frac{r}{2},since |\sin 2\theta| \leq 1. As (x,y) \to (0,0), r \to 0 regardless of \theta, so -\frac{r}{2} \leq \frac{xy}{\sqrt{x^2 + y^2}} \leq \frac{r}{2} squeezes the function to 0, confirming the limit.[31] This coordinate transformation highlights how the squeeze theorem verifies path-independence in multivariable settings.[5]
Applications and Generalizations
In Calculus and Analysis
The squeeze theorem is frequently employed in calculus to evaluate limits that result in indeterminate forms, such as $0/0 or \infty/\infty, by establishing inequalities that bound the target function between two others whose limits are known. For instance, consider the limit \lim_{x \to 0} \frac{\sin x}{x}, which is an indeterminate form $0/0; geometric arguments on the unit circle yield the inequality \cos x \leq \frac{\sin x}{x} \leq 1 for $0 < x < \pi/2, and taking limits as x \to 0^+ gives both bounds approaching 1, so the limit is 1 by the squeeze theorem.[32] Similarly, for \lim_{x \to 0} x^2 \sin(1/x), another $0/0 form involving oscillation, the inequality -x^2 \leq x^2 \sin(1/x) \leq x^2 holds since |\sin(1/x)| \leq 1, and both -x^2 and x^2 approach 0, yielding the limit 0.[32]In the context of derivatives, the squeeze theorem underpins proofs for trigonometric functions, particularly the derivative of \sin x at x = 0. The difference quotient is \frac{\sin x - \sin 0}{x - 0} = \frac{\sin x}{x}, whose limit as x \to 0 is 1 as established above, confirming \frac{d}{dx} \sin x \big|_{x=0} = \cos 0 = 1. This result extends to the general derivative \frac{d}{dx} \sin x = \cos x via the chain rule and limit laws, with the squeeze theorem providing the foundational limit.[33]For improper integrals, the squeeze theorem aids in assessing convergence by bounding integrands or partial integrals. A classic example is the Dirichlet integral \int_1^\infty \frac{\sin x}{x} \, dx, which converges conditionally; over each interval [n\pi, (n+1)\pi], the absolute value of the integral is bounded above by \int_{n\pi}^{(n+1)\pi} \frac{|\sin x|}{x} \, dx \leq \frac{1}{n\pi} \int_{n\pi}^{(n+1)\pi} |\sin x| \, dx = \frac{2}{n\pi}, and the series of these bounds behaves like the harmonic series, but the alternating signs ensure convergence via the Dirichlet test, with squeeze providing the term bounds.[34]The squeeze theorem also proves continuity for functions exhibiting oscillations, such as f(x) = x^2 \sin(1/x) for x \neq 0 and f(0) = 0. To verify continuity at 0, compute \lim_{x \to 0} f(x); since -1 \leq \sin(1/x) \leq 1, it follows that -x^2 \leq x^2 \sin(1/x) \leq x^2, and both bounds approach 0 as x \to 0, so the limit is 0 by the squeeze theorem, matching f(0).[23]As a discrete analog in series, the comparison test leverages squeeze-like bounding: if $0 \leq a_n \leq b_n for all n sufficiently large and \sum b_n converges, then \sum a_n converges, mirroring the theorem's logic for sequences of partial sums.[35]
Extensions to One-Sided and Multivariable Limits
The squeeze theorem extends naturally to one-sided limits. For the right-hand limit as x \to a^+, if there exists \delta > 0 such that f(x) \leq g(x) \leq h(x) for all x \in (a, a + \delta), and \lim_{x \to a^+} f(x) = \lim_{x \to a^+} h(x) = L, then \lim_{x \to a^+} g(x) = L. A similar statement holds for the left-hand limit as x \to a^-, with the inequality holding on (a - \delta, a). This adaptation preserves the core idea of bounding the function between two others whose one-sided limits agree, allowing evaluation of indeterminate forms from one side.[36][23]A representative example is \lim_{x \to 0^+} \sqrt{x} \sin(1/x). Since -1 \leq \sin(1/x) \leq 1, it follows that -\sqrt{x} \leq \sqrt{x} \sin(1/x) \leq \sqrt{x} for x > 0. As x \to 0^+, both -\sqrt{x} \to 0 and \sqrt{x} \to 0, so by the one-sided squeeze theorem, the limit is 0. This illustrates how the theorem handles oscillatory behavior near a boundary point from one direction.In multivariable calculus, the squeeze theorem generalizes to functions of several variables. If g(\mathbf{x}) \leq f(\mathbf{x}) \leq h(\mathbf{x}) for all \mathbf{x} sufficiently close to \mathbf{a} \in \mathbb{R}^n (excluding \mathbf{a} itself), and both \lim_{\mathbf{x} \to \mathbf{a}} g(\mathbf{x}) and \lim_{\mathbf{x} \to \mathbf{a}} h(\mathbf{x}) exist and equal L, then \lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = L. However, for this to hold, the limits of g and h must be path-independent, as multivariable limits require agreement along all paths approaching \mathbf{a}; path dependence in g or h would invalidate the conclusion. This extension is particularly useful for confirming limits in higher dimensions where direct computation is challenging.[37][29][38]Further generalization to metric spaces appears in the context of sequences, where if \{r_n\} is a null sequence in \mathbb{R} and \{x_n\} is a sequence in the metric space (S, d) satisfying d(p, x_n) \leq r_n for all n, then x_n \to p. This form leverages distance bounds rather than ordering, providing a tool analogous to the squeeze theorem for convergence without relying on total orders, and it supports arguments involving uniform continuity by controlling distances uniformly near a point.[39]