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Question
Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:
f(x) `{:(= x^2 + x - 3,"," "for" x ∈ [ -5, -2)),(= x^2 - 5,"," "for" x ∈ (-2, 5]):}`
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Solution
f is continuous in [– 5, – 2) and in (– 2, 5] since it is a polynomial function.
Continuity at x = – 2
f(x) = x2 + x – 3, for x ∈ [– 5, – 2)
∴ `lim_(x -> - 2^-) "f"(x) = lim_(x -> -2) (x^2 + x - 3)` = 4 – 2 – 3 = – 1
Also, f(x) = x2 + 5, for x ∈ [– 2, 5)
∴ `lim_(x -> -2^+) "f"(x) = lim_(x -> - 2) (x^2 - 5)` = 4 – 5 = – 1
∴ `lim_(x -> -2^-) "f"(x) = lim_(x -> - 2^+) "f"(x)` = – 1
∴ `lim_(x -> - 2) "f"(x)` = – 1
But f(– 2) is not defined.
∴ f is discontinuous at x = – 2
This discontinuity is removable and can be removed by redefining the function as follows:
f(x) `{:(= x^2 + x - 3, "," "for" x ∈ [ -5, -2)),(= x^2 - 5, "," "for" x ∈ (-2, 5]),(= -1, "," "for" x = -2):}`
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