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Question
Test the continuity of the following function at the point or interval indicated against them :
f(x) `{:(= 4x + 1",", "for" x ≤ 8/3),(= (59 - 9x)/3 ",", "for" x > 8/3):}} "at" x = 8/3`
Solution
`lim_(x -> 8^-/3) "f"(x) = lim_(x -> 8^-/3) (4x + 1)`
= `4 (8/3) + 1`
= `35/3`
`lim_(x -> 8^+/3) "f"(x) = lim_(x -> 8^+/3) ((59 - 9x)/3)`
= `(59 - 9(8/3))/3`
= `(59 - 24)/3`
= `35/3`
f(x) = 4x + 1
∴ `"f"(8/3) = 4(8/3) + 1 = 35/3`
∴ `lim_(x -> 8^-/3) "f"(x) = lim_(x -> 8^+/3) "f"(x) = "f"(8/3)`
f(x) is continuous at x = `8/3`.
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