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