Understanding Continuity of the Function f(x) frac{x^2 - 1}{x - 1} at x -1

Introduction

In this article, we will explore the continuity of the function f(x) frac{x^2 - 1}{x - 1} at x -1. Understanding the continuity of a function is crucial in calculus and analysis, as it helps us determine whether a function behaves consistently around a specific point in its domain. We will systematically check the three conditions for continuity at a point and conclude whether the function is continuous at x -1.

Let's begin by evaluating the function and its limit as x approaches -1.

Evaluating the Function at x -1

First, we need to evaluate f(-1).

f(-1) frac{(-1)^2 - 1}{-1 - 1} frac{1 - 1}{-2} frac{0}{-2} 0

However, this is incorrect because the function is undefined at x -1, as we would have a division by zero.

Finding the Limit as x Approaches -1

To find the limit, we first simplify the function:

f(x) frac{x^2 - 1}{x - 1} frac{(x 1)(x - 1)}{x - 1}

For x ≠ -1, we can cancel the common factor x - 1:

f(x) x 1

Next, we evaluate the limit of f(x) as x approaches -1:

lim_{x to -1} f(x) lim_{x to -1} (x 1) -1 1 0

Checking Continuity

To check the continuity, we need to verify the following conditions:

1. The function is defined at x -1 f(x) frac{x^2 - 1}{x - 1} is undefined at x -1 because we have a division by zero.

2. The limit of the function as x approaches -1 exists As calculated, the limit of f(x) as x approaches -1 exists and is equal to 0.

3. The limit equals the function value at that point

From the evaluation, we see that the limit as x approaches -1 is 0, but the function is undefined at x -1.

Since the function is undefined at x -1, the function cannot be continuous at x -1.

Conclusion

The function f(x) frac{x^2 - 1}{x - 1} is not continuous at x -1.

Alternative Interpretations

There are alternative interpretations of the expression, which can be evaluated for continuity:

1. If the expression was meant to be f(x) frac{x^2 - 1}{x 1}, then the function is undefined at x -1 and not continuous at that point.

2. If the expression was meant to be f(x) frac{x^2 - 1}{x - 1}, then simplifying gives f(x) x 1, which is continuous at all real numbers, including x -1.

3. If the expression was meant to be f(x) frac{x^2 - 1}{x - 1}, then the function is not defined at x -1, but we can define the function to be f(x) begin{cases} -2 text{if } x -1 x 1 text{if } x ≠ -1 end{cases} to make it continuous at x -1.