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Question
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) `{:(= x^2 - 3x - 2",", "for" x < -3),(= 3 + 8x",", "for" x > -3):}`
Solution
f(x) `{:(= x^2 - 3x - 2",", x < -3),(= 3 + 8x",", x > -3):}`
f(x) is a polynomial function for both the intervals.
∴ f(x) is continuous for both the given intervals.
Let us test the continuity at x = − 3
`lim_(x -> -3^-) "f"(x) = lim_(x -> -3^-) (x^2 - 3x - 2)`
= (− 3)2 − 3(− 3) − 2
= 9 + 9 − 2
= 16
`lim_(x -> -3^+) "f"(x) = lim_(x -> -3^+) (3 + 8x)`
= 3 + 8(− 3)
= − 21
∴ `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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