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Question
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) `{:(= 4 + sin x",", "for" x < pi),(= 3 - cos x",", "for" x > pi):}`
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Solution
`lim_(x -> pi^-) "f"(x) = lim_(x -> pi) (4 + sin x)`
= 4 + sin π
= 4 + 0
= 4
`lim_(x -> pi^+) "f"(x) = lim_(x -> pi) (3 - cos x)`
=3 – cos π
=3 – (– 1)
= 4
∴ `lim_(x -> pi^-) "f"(x) = lim_(x -> pi^+) "f"(x)` = 4
∴ `lim_(x -> pi) "f"(x)` exists and equals 4
But f(π) is not defined
If we define f(π) = 4, f will be continuous at x = π
∴ the discontinuity is removable.
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