Continuity of Exponential Functions at x 0

Exponential functions are fundamental in mathematics, playing crucial roles in various domains such as calculus, physics, and engineering. Specifically, the functions fx ex and fx e-x are particularly important. In this article, we will explore the continuous nature of these functions at x 0, ensuring that the content is optimized for Google's search algorithms and adheres to SEO best practices.

Introduction to Continuity

In mathematics, a function f is said to be continuous at a point x a if the following three conditions are satisfied:

The function f must be defined at x a: This means that the function does not have any undefined values at the point of interest. The limit of the function as x approaches a exists: This implies that the function approaches a specific value as it gets closer to a. The limit of the function as x approaches a is equal to the value of the function at x a: Mathematically, this can be expressed as limx→a f(x) f(a).

Continuity of fx ex at x 0

Let's begin with the function fx ex.

Step 1: Evaluating the Function at x 0

First, we need to assess whether the function is defined at x 0: [text{f(0) e}^0 1]

Step 2: Finding the Limit as x Approaches 0

Next, we need to evaluate the limit as x approaches 0: [lim_{x to 0} e^x e^0 1]

Since both the function value at x 0 and the limit as x approaches 0 are equal to 1, it satisfies the continuity condition for fx ex at x 0.

Continuity of fx e-x at x 0

Now let's consider the function fx e-x.

Step 1: Evaluating the Function at x 0

We start by evaluating the function at x 0: [text{f(0) e}^{-0} 1]

Step 2: Finding the Limit as x Approaches 0

Next, we need to evaluate the limit as x approaches 0: [lim_{x to 0} e^{-x} e^0 1]

Since both the function value at x 0 and the limit as x approaches 0 are equal to 1, the function fx e-x is also continuous at x 0.

Proof by Definition

To further substantiate the continuity of fx ex and fx e-x at x 0, we will provide a step-by-step proof based on the definition of limits.

Proof for fx ex

Function Definition at x 0 [text{f(0) e}^0 1] Limit as x Approaches 0 [lim_{x to 0} e^x e^0 1]

Since the limit and the function value at x 0 are equal, the function is continuous at x 0.

Proof for fx e-x

Function Definition at x 0 [text{f(0) e}^{-0} 1] Limit as x Approaches 0 [lim_{x to 0} e^{-x} e^0 1]

Again, since both the limit and the function value at x 0 are equal, the function is continuous at x 0.

Conclusion

Both the functions fx ex and fx e-x are proven to be continuous at x 0 based on the limit definition of continuity. This proof can be summarized as follows:

The function values at x 0 are well-defined and evaluated to 1. The limits as x approaches 0 for both functions are also 1, verifying the second criterion of continuity.

In summary, both exponential functions are continuous at x 0, meeting the necessary criteria for continuity.