Using the Squeeze Theorem to Evaluate Limits in Calculus

Squeeze Theorem

The Squeeze Theorem is a powerful tool in calculus that allows you to evaluate limits when direct substitution or other basic techniques don’t work

The Squeeze Theorem is a powerful tool in calculus that allows you to evaluate limits when direct substitution or other basic techniques don’t work. It is also known as the Sandwich Theorem or the Pinching Theorem.

The statement of the Squeeze Theorem is as follows: Let f(x), g(x), and h(x) be functions defined on an interval containing a point ‘c’, except possibly at ‘c’ itself. Suppose that for all x in the interval, except possibly at ‘c’, we have f(x) ≤ g(x) ≤ h(x). If the limit of f(x) and h(x) as x approaches ‘c’ is L, then the limit of g(x) as x approaches ‘c’ is also L.

To understand this theorem better, let’s consider an example. Suppose we want to find the limit as x approaches 0 of the function f(x) = x^2*sin(1/x). This function oscillates wildly around 0, making it difficult to directly evaluate the limit. However, we can apply the squeeze theorem to help us out.

We know that for all x in the domain of f(x), -1 ≤ sin(1/x) ≤ 1. Multiplying this inequality by x^2, we get -x^2 ≤ x^2*sin(1/x) ≤ x^2.

Now, let’s evaluate the limits of the squeezed functions as x approaches 0. The limit of -x^2 as x approaches 0 is 0, and the limit of x^2 as x approaches 0 is also 0. Therefore, by the Squeeze Theorem, the limit of x^2*sin(1/x) as x approaches 0 is also 0.

In summary, the Squeeze Theorem allows us to evaluate limits by sandwiching the given function between two easier-to-evaluate functions, whose limits are already known. It provides a useful tool for finding limits in situations where direct evaluation is not possible or straightforward.

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