Question

What type of discontinuity does the function -x^2 + 2x + 2 have at x = 3, given that the function is defined as 0 when x equals 3, and it is equal to -x^2 + 2x + 2 for all other x values except x doesn't equal 3? Choose the correct description from the options A, B, C, D, or E: A) Removable discontinuity B) Infinite discontinuity C) Continuous D) Oscillating discontinuity E) None of these.

Answer 1

The function has a removable discontinuity at x = 3, because it has a defined limit that does not match the actual value at that point.

Step-by-step explanation:

The function given is -x^2 + 2x + 2, except at x = 3, where the function is defined to be 0. If we substitute x = 3 into the function, we would normally get -3^2 + 2*3 + 2 = -9 + 6 + 2 = -1. However, the function is redefined to be 0 at x = 3. This creates a situation where the limit of the function as x approaches 3 is -1, but the actual value of the function at x = 3 is 0.

This type of discontinuity, where the limit exists but is not equal to the function's value at the point of discontinuity, is known as a removable discontinuity.