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Question
Test the continuity of the following function at the points indicated against them:
f(x) = 4x + 1, for x ≤ 3
= `(59 - 9x)/3`, for x > 3 at x = `8/3`.
Solution
`lim_(x→(8/3)^-) "f"(x) = lim_(x→(8/3)^-) (4x + 1)`
= `4(8/3) + 1`
= `32/3 + 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, x ≤ `(8/3)`
∴ `"f"(8/3) = 4(8/3) + 1`
= `32/3 + 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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