A function f(x) is said to have a removable discontinuity at x=a if:

Question

1. f is either not defined or not continuous at x = a

2. f(a) could either be defined or redefined so that the new function is continuous at x = a

Let f(x) = { 7÷x + -6x+7 ÷ x(x-1) if x ≠ 0,1

{ 6 if x ≠ 0

Show that f(x) has a removable discontinuity at x = 0 and determine what value for f(0) would make f(x)  continuous at x=0

Must redefine f(0) =

ps. The discontinuity at x = 1 is not a removable discontinuity, just in case you were wondering.

Answer 1

You can combine the first and second halves of the equation so they have a common denominator by multiplying the left hand side by (x-1). Then the equation looks like:

f(x) = (7(x-1) + (-6x) + 7) / (x)(x-1)

Simplify the top half of the equation. Then you can eliminate division by x if you multiply the top & bottom halves of the equation by x. Note that there is still an (x-1) in the denominator afterwards.