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Question
Discuss the continuity of the following function at the point or on the interval indicated against them. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous:
f(x) `{:(= x^2 + 2x + 5"," , "for" x ≤ 3),( = x^3 - 2x^2 - 5",", "for" x > 3):}`
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Solution
`lim_(x -> 3^-) "f"(x) = lim_(x -> 3^-) (x^2 + 2x + 5)`
= (3)2 + 2(3) + 5
= 9 + 6 + 5
= 20
`lim_(x -> 3^+) "f"(x) = lim_(x -> 3^+) (x^3 - 2x^2 - 5)`
= (3)3 + 2(3)2 – 5
= 27 – 18 – 5
= 4
∴ `lim_(x -> 3^-) "f"(x) ≠ lim_(x -> 3^+) "f"(x)`
∴ `lim_(x -> 3) "f"(x)` does not exist
∴ f(x) is discontinuous at x = 3
This continuity is irremovable.
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