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The Squeeze Theorem, also known as the Sandwich Theorem, is a useful tool in calculus for finding the limit of a function. It works when you have a function that is "squeezed" between two other functions whose limits are known and equal at a particular point. Formally, if you have three functions \(f(x)\), \(g(x)\), and \(h(x)\) such that \(f(x) \leq g(x) \leq h(x)\) 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 both \(L\), then the limit of \(g(x)\) as \(x\) approaches \(c\) is also \(L\). In mathematical notation: $$ \text{If } \lim_{{x \to c}} f(x) = \lim_{{x \to c}} h(x) = L \text{ and } f(x) \leq g(x) \leq h(x) \text{ for all } x \text{ near } c, \text{ then } \lim_{{x \to c}} g(x) = L. $$ Here's an intuitive way to think about it: Imagine you have three runners on a track. Runner \(f(x)\) and Runner \(h(x)\) are both heading towards the finish line at the same pace and will arrive there together. Runner \(g(x)\) is always running between them. Because \(g(x)\) is squeezed between \(f(x)\) and \(h(x)\), \(g(x)\) has no choice but to also reach the finish line at the same time as the other two. An example: Consider \(f(x) = -x\), \(g(x) = \frac{\sin(x)}{x}\), and \(h(x) = x\). Near \(x = 0\): $$ -x \leq \frac{\sin(x)}{x} \leq x $$ As \(x\) approaches 0, both \(\lim_{{x \to 0}} (-x) = 0\) and \(\lim_{{x \to 0}} x = 0\). So by the Squeeze Theorem, \(\lim_{{x \to 0}} \frac{\sin(x)}{x} = 0\). I hope this helps clarify the Squeeze Theorem!
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Sun, 07 Jul 2024 05:48 GMT