Squeeze Theorem

The Squeeze Theorem is a powerful tool in calculus that allows us to evaluate limits of functions that are "sandwiched" between two other functions.

Table of Contents

• Table of Contents
• Introduction

Introduction

To understand the Squeeze Theorem, let's consider the following scenario: you are working with a list of items to deliver to a customer. These items are sorted by how heavy they are. Let's say you want to know the weight of the fifth item in the list, but you don't know the exact weight of the item. However, you do know that the fourth item weighs 10 kg and the sixth item weighs 10 kg as well. Well, you can conclude that the weight of the fifth item must be 10 kg as well, since it is "sandwiched" between the fourth and sixth items.

The Squeeze Theorem works in a similar way. If you have a function that is "sandwiched" between two other functions and , and the limits of and as approaches a certain value are equal, then the limit of as approaches the same value must also be equal to that limit. We shall now explore this concept in more detail with an example:

Recall that when we tried to evaluate this limit using direct substitution, we encountered an indeterminate form of . It's also not immediately clear how we can algebraically manipulate the expression to evaluate the limit (although it is possible, say by using the Taylor series expansion of ). However, we can use the Squeeze Theorem to evaluate this limit.

We will need to find two functions that "sandwich" and whose limits as approaches are known. To do this, we can make use of some geometry and trigonometry.