Question

How would you "remove the discontinuity" of f? In other words, how would you define f(5) in order to make f continuous at 5? f(x) = [(x^2 -2x -15)/x-5]

Answer 1

Given f(x) = [(x^2 -2x -15)/x-5], notice it is a rational function

But the numerator x^2 -2x -15 could be factored, which yields

(x – 5)(x + 3)

Therefore f(x) = (x – 5)(x + 3)/ (x-5)

Cancelling x-5, f(x)  = x + 3

In this way, the f(x) is continuous at any point, and is basically a line.