Proving the Continuity of a Function at a Point: fx x2 at x 4

Introduction

Understanding the continuity of a function at a specific point is a fundamental concept in calculus. In this article, we will explore how to prove that the function fx x2 is continuous at the point x 4. We will cover the necessary steps and provide examples to help you grasp the concept thoroughly.

What Does Continuity Mean?

A function fx is continuous at a point x a if it meets three conditions:

It is defined at x a. The limit of fx as x approaches a exists. The limit of fx as x approaches a is equal to the value of the function at x a.

Step-by-Step Proof for Continuity at x 4

1. Define the Function at x 4

To prove continuity, we first need to establish that the function fx x2 is defined at x 4.

fx x2 is a polynomial function, and all polynomial functions are defined for all real numbers. Therefore, f4 exists and is equal to 42 16.

2. Existence of the Limit as x Approaches 4

Next, we need to show that the limit of fx as x approaches 4 exists. This means that as x gets closer to 4 from both the left and the right, the value of fx gets closer to 16.

Let's verify this:

When x is slightly less than 4, say x 3.99999, then fx 3.999992 ≈ 15.9999996000009. When x is slightly greater than 4, say x 4.00001, then fx 4.000012 ≈ 16.000040000001.

As shown, the values of fx get arbitrarily close to 16, confirming the existence of the limit.

3. Limit of fx as x Approaches 4 is Equal to f4

The final step is to show that the limit of fx as x approaches 4 is equal to the value of the function at x 4.

We have already established that lim x - 4 fx 16 and f4 42 16. Therefore, lim x - 4 fx f4.

Since A C and B C imply that A B, we have proven that lim x - 4 fx f4.

Verification Using a Table

A table can also help verify that fx is defined at x 4 and that the function approaches 16 as x approaches 4 from both sides.

xfx x2 3.915.21 3.9915.9201 3.99915.992001 3.999915.99920001 4.000116.00040001 4.00116.008001 4.0116.0801 4.116.81

Delta-Epsilon Proof (Optional Content)

For a more rigorous proof, the delta-epsilon definition of a limit can be used. This method involves showing that for every ? 0, there exists a δ 0 such that for all x with 0 |x - 4| δ, it follows that |fx - 16| ?.

For the function fx x2, the proof would go as follows:

Select an arbitrary ? 0. Choose δ min{1, ?/7}. For any x such that 0 |x - 4| δ, we have:

|fx - 16| |x2 - 16| |(x - 4)(x 4)| 7δ 7(?/7) ?

This confirms that lim x - 4 fx 16.

Conclusion

By verifying the three conditions for continuity and using the delta-epsilon method, we have shown that the function fx x2 is continuous at the point x 4. Understanding and proving continuity is crucial for advanced calculus, and this method provides a solid foundation for further studies in mathematical analysis.