SqueezeTheorem

The Squeeze Theorem, also known as the Sandwich Theorem, is a fundamental concept in calculus used to determine the limit of a function by comparing it to two other functions whose limits are known. It is particularly useful when the direct evaluation of a limit is difficult or indeterminate.

The theorem states that if three functions, f(x), g(x), and h(x), satisfy the inequalities f(x) ≤ g(x) ≤ h(x) for all x in some interval around a point c (except possibly at c itself), and if the limits of f(x) and h(x) as x approaches c are equal to the same value L, then the limit of g(x) as x approaches c must also be L. Mathematically, this is expressed as:

If lim(x→c) f(x) = L, lim(x→c) h(x) = L, and f(x) ≤ g(x) ≤ h(x) near c (except possibly at c), then lim(x→c) g(x) = L.

A common application of the Squeeze Theorem involves trigonometric functions. For example, when evaluating lim(x→0) x sin(1/x), direct substitution leads to an indeterminate form (0 × ∞). However, since -1 ≤ sin(1/x) ≤ 1 for all x, multiplying through by x (which is positive near 0) gives -x ≤ x sin(1/x) ≤ x. Taking the limit as x approaches 0, both -x and x converge to 0, so by the Squeeze Theorem, lim(x→0) x sin(1/x) = 0.

The theorem relies on the idea that if a function is "squeezed" between two other functions that converge to the same limit, it must also converge to that same limit. This approach is especially helpful in cases where the behavior of the function is bounded by simpler functions whose limits are easier to determine. The Squeeze Theorem is widely used in analysis and mathematical proofs to establish the existence of limits.