Understanding the Boundedness Theorem and Its Role in Limit Calculations

An Introduction to the Boundedness Theorem

The Boundedness Theorem is an important principle in the analysis of limits that helps us examine the behavior of a function near a specific point.
This theorem states that if a function is bounded in a neighborhood of a point, we can use this property to analyze its limit.

The Formal Statement of the Boundedness Theorem

If f(x) is bounded on an open interval containing the point a (except possibly at a itself), meaning:

|f(x)| ≤ M

for some constant M, then this property can be used to analyze the limit of expressions involving f(x).

The Importance of the Boundedness Theorem

This theorem is especially useful in the following situations:

• Analyzing oscillatory functions that do not have a fixed value
• Studying limits of functions that involve products or combinations of other functions
• Simplifying calculations in challenging limit problems
• Supporting the proof of the Squeeze Theorem and other limit theorems

Example 1: Boundedness of sin(x)

We know that:

|sin(x)| ≤ 1

This means sin(x) is a bounded function.
Now consider the function:

f(x) = x · sin(x)

Using the boundedness of sin(x):

|x · sin(x)| ≤ |x|

And since:

lim (x → 0) |x| = 0

we conclude:

lim (x → 0) x · sin(x) = 0

Example 2: Boundedness of cos(1/x)

Consider the function:

f(x) = x² cos(1/x)

We know that:

|cos(1/x)| ≤ 1

Therefore:

|x² cos(1/x)| ≤ x²

And since:

lim (x → 0) x² = 0

we conclude:

lim (x → 0) x² cos(1/x) = 0

The Relationship Between the Boundedness Theorem and the Squeeze Theorem

In many cases, the Boundedness Theorem serves as a foundation for applying the Squeeze Theorem.
When a function is bounded, it can be placed between two other functions, allowing us to use the Squeeze Theorem to determine its limit.

Conclusion

The Boundedness Theorem is a key tool in the analysis of limits.
By providing a simple framework for controlling the behavior of functions, it makes the calculation of limits in many problems much easier.
Understanding this theorem is essential for success in advanced topics such as mathematical analysis, continuity, and differentiation.