Question

In your own words, explain the Squeeze Theorem.

Short answer

Squeeze theorem is a concept in calculus that is used to estimate the limits of functions. If a function \(f(x)\) is trapped between function \(g(x)\) and \(h(x)\), and both \(g(x)\) and \(h(x)\) approach a common limit at a certain point, then \(f(x)\) will also approach that same limit.

Step-by-step solution

Define the Squeeze Theorem

To begin with, Squeeze Theorem is a popular mathematical principle that is used in Calculus to evaluate limiting behaviour of certain functions. It is defined as: If a function \(f\) is such that for all \(x\) in an interval, we have \(g(x) <= f(x) <= h(x)\) and the limit of \(g\) and \(h\) as \(x\) approaches a certain value \(a\) is \(l\), then the limit of the function \(f(x)\) as \(x\) approaches \(a\) is also \(l\). In simpler words, if \(f(x)\) is 'squeezed' by \(g(x)\) and \(h(x)\) and both \(g(x)\) and \(h(x)\) approach a common limit at a certain point, then \(f(x)\) must also approach that same limit.

Explain the Implication of Squeeze Theorem

The Squeeze Theorem is primarily utilized for understanding the behavioral tendencies of a function as it approaches a specific value. It allows us to ascertain the limit of difficult functions by 'squeezing' them between two simpler functions whose limits are known or easily found.

Provide an Example for Clarification

As an example, consider three functions \(f(x) = x^2, g(x) = 2x\) and \(h(x) = 3x\). Here, for all \(x > 0\), \(x^2 <= 2x <= 3x\), that means \(f(x)\) is squeezed between \(g(x)\) and \(h(x)\). As \(x\) approaches 1, the limit of both \(g(x)\) and \(h(x)\) is either 2 or 3 respectively. Therefore, via the Squeeze theorem, we infer that as \(x\) approaches 1, the limit of \(f(x)\) should also be between 2 and 3.