Removable Discontinuities in Functions: Explanation and Example using f(x) = (x^2 – 4)/(x – 2)

Removable discontinuity

When a non-permissible value in the denominator has a common factor in the numerator

A removable discontinuity is a type of discontinuity in a function where the function is undefined at a certain point, but the point can be removed and replaced with a defined value, resulting in a continuous function. This type of discontinuity occurs when the function approaches a specific value at a certain point, but does not equal that value at that point.

To illustrate this concept, let’s take the example of the function f(x) = (x^2 – 4)/(x – 2). At x = 2, this function is undefined because dividing by zero is undefined. However, if we simplify the function by factoring the numerator, f(x) can be written as f(x) = (x + 2)(x – 2)/(x – 2). Now we see that the factor (x – 2) cancels out, leaving f(x) = x + 2. Therefore, as x approaches 2, the function f(x) approaches 4.

In this case, we have a removable discontinuity at x = 2 because the function is undefined at that point, but can be continuously defined by removing the singularity and introducing the value 4 at that point.

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