Question

Which of the following functions f has a removable discontinuity at x = x₀? If the discontinuity is removable, find a function g that agrees with f for x ≠ x₀ and is continuous on R.

`f(x) = (x^2 - 2x - 8)/(x + 2), x_0` = – 2

Answer 1

f(x) is not defined at x = – 2

`lim_(x -> -2) (x^2 - 2x - 8)/(x + 2) =  lim_(x -> - 2) (x^2 - 4x + 2x - 8)/(x + 2)`

= `lim_(x -> -2) x(x(x - 4) + 2(x - 4))/(x + 2)`

= `lim_(x -> - 2) ((x + 2)(x - 4))/(x + 2)`

= `lim_(x -> - 2) (x - 4)`

= – 2 – 4

= – 6

∴ `lim_(x -> -2) (x^2 - 2x - 8)/(x + 2)` exists.

Redefine the function f(x) as

`g(x) = {{:((x^2 - 2x - 8)/(x + 2),  "if"  x ≠ - 2),(-6,  "if"  x = - 2):}` 

∴ f(x) has a removable discontinuity at x = – 2.

Clearly, g(x) is continuous on R.