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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 + 5x + 1"," , "for" 0 ≤ x ≤ 3),(= x^3 + x + 5"," , "for" 3 < x ≤ 6):}`
Solution
`lim_(x -> 3^-) "f"(x) = lim_(x -> 3^-) (x^2 + 5x + 1)`
= `lim_(x -> 3^-) (3)^2 + 5(3) + 1`
= 9 + 15 + 1
= 25
`lim_(x -> 3^+) "f"(x) = lim_(x -> 3^+) (x^3 + x + 5)`
= (3)3 + 3 + 5
= 35
∴ `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
∴ f(x) has a jump discontinuity at x = 3
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