The Squeeze Theorem: Evaluating Limits by Comparing Functions

Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem or the Pinching Theorem, is a powerful tool used in calculus to evaluate the limit of a function by comparing it to two other functions whose limits are known

The Squeeze Theorem, also known as the Sandwich Theorem or the Pinching Theorem, is a powerful tool used in calculus to evaluate the limit of a function by comparing it to two other functions whose limits are known.

The statement of the Squeeze Theorem can be described as follows:

Suppose that for all x in some interval, except possibly at a point c (which could be the endpoint of the interval), we have f(x) ≤ g(x) ≤ h(x), and also assume that the limits of f(x) and h(x) as x approaches c exist and are equal to some common value L. Then, the limit of g(x) as x approaches c also exists and is equal to L.

Visually, we can think of the function g(x) being “squeezed” between the functions f(x) and h(x), as shown below:

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–|———–|——–|–> x
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f(x) g(x) h(x)

Essentially, the Squeeze Theorem allows us to establish the limit of a function by comparing it to other functions that are known to approach the same limit. It is particularly useful when dealing with functions that are not easily evaluated directly.

To apply the Squeeze Theorem, we need to find two other functions, f(x) and h(x), that “squeeze” the function g(x) and whose limits are known. By establishing that g(x) lies between f(x) and h(x), and that the limits of f(x) and h(x) are equal to L, we can conclude that the limit of g(x) is also L.

Let’s consider an example to illustrate the use of the Squeeze Theorem:

Example:
Find the limit of f(x) = x^2 cos(1/x) as x approaches 0.

To solve this, we need to find two other functions, f(x) and h(x), that “squeeze” f(x) and whose limits are known. Since cosine function is bounded between -1 and 1 (i.e., -1 ≤ cos(1/x) ≤ 1), we can set f(x) = x^2(-1) and h(x) = x^2(1).

Now, we can simplify the inequalities using the known properties of limits:

– x^2 ≤ x^2 cos(1/x) ≤ x^2 (since -1 ≤ cos(1/x) ≤ 1 for all x)

Taking the limit as x approaches 0 for all terms, we find:

– lim(x→0)(x^2) ≤ lim(x→0)(x^2 cos(1/x)) ≤ lim(x→0)(x^2)
– 0 ≤ lim(x→0)(x^2 cos(1/x)) ≤ 0

Hence, the limit of f(x) = x^2 cos(1/x) as x approaches 0 is 0. The Squeeze Theorem allows us to conclude that the function f(x) approaches 0 as x approaches 0, even though it is not directly evaluable at x = 0.

The Squeeze Theorem is a valuable tool in calculus that allows us to evaluate limits by comparing them to known functions. It is especially useful for functions that are not easily evaluated directly or when direct application of other limit theorems is not possible.

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