Is the Function f(x) 1/x - 1 Continuous at x 1?

In this article, we will explore whether the function f(x) 1/x - 1 is continuous at x 1. To understand the continuity of a function at a specific point, we need to evaluate three conditions:

1. The Function is Defined at x 1

For the function to be continuous at a point, it must be defined at that point. Let's substitute x 1 into the function:

f(1) 1/1 - 1 0

2. The Limit of the Function as x Approaches 1 Exists

The limit of the function as x approaches a point can be evaluated by substituting that point into the function if it is defined and the function is well-behaved. Let's evaluate the limit as x approaches 1:

limx → 1 f(x) limx → 1 (1/x - 1) 1/1 - 1 0

3. The Limit Equals the Function Value at That Point

For the function to be continuous at x 1, the limit of the function as x approaches 1 (which we just calculated) must equal the value of the function at x 1. In this case:

limx → 1 f(x) f(1) 0

Since all three conditions are satisfied, the function f(x) 1/x - 1 is continuous at x 1.

Common Misconceptions and Examples

It might be tempting to think that the function is discontinuous at x 1 because of the form 1/x - 1. To clarify, let's consider two different representations of the function:

Example 1: f(x) 1/x - 1

If we write the function as f(x) 1/x - 1, it is indeed continuous at x 1 through the steps outlined above.

Example 2: f(x) 1/(x-1)

However, if the function is written as f(x) 1/(x-1), the situation changes:

If f(x) 1/(x-1), then:

limx → 1 f(x) limx → 1 1/(x-1) 1/0 (which is undefined)

Since the limit does not exist (or is undefined) at x 1, the function is discontinuous at x 1.

Conclusion

Whether a function is continuous at a specific point depends on how the function is defined and the behavior of the function around that point. The function f(x) 1/x - 1 satisfies the conditions for continuity at x 1 if it is written in the form 1/x - 1.