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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
Solution
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.
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