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^3 + 64)/(x + 4), x_0` = – 4

Answer 1

The function f(x) is not defined at x = – 4.

`f(x) = (x^3 + 4^3)/(x + 4)`

`f(x) = ((x + 4)(x^2 - 4x + 16))/(x + 4)`

`f(x) = x^2 - 4x + 16`

`lim_(x -> - 4) f(x) =  lim_(x -> - 4) (x^2 - 4x + 16)`

= `(- 4)^2 - 4 xx - 4 + 16`

= 16 + 16 + 16

`lim_(x -> - 4) f(x)` = 48

 Limit the function f(x) exist at x = – 4.

∴ The function f(x) has a removable discontinuity at x = – 4.

Redefine the function f (x) as

`g(x) = {{:((x^3 + 64)/(x + 4),  "if"  x ≠ - 4),(48,  "if"  x = - 4):}`

Clearly, the function g(x) is continuous on R.