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Question
The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it become continuous :
f(x) `{:(= 3x + 2",", "for" -4 ≤ x ≤-2),(= 2x - 3";", "for" -2 < x ≤ 6):}`
Solution
f(x) `{:(= 3x + 2",", "for" -4 ≤ x ≤-2),(= 2x - 3";", "for" -2 < x ≤ 6):}`
`lim_(x -> -2^-) "f"(x) = lim_(x -> -2^-) (3x + 2)`
= 3(– 2) + 2
= – 4
`lim_(x -> -2^+) "f"(x) = lim_(x -> -2^+) (2x - 3)`
= 2(– 2) – 3
= – 7
∴ `lim_(x -> -2^-) "f"(x) ≠ lim_(x -> -2^+) "f"(x)`
∴ `lim_(x -> -2^-) "f"(x)` does not exist
∴ f(x) is discontinuous at x = – 2
This discontinuity is irremovable.
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