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Question
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) `{:(= (x^2 - 3x - 10)/(x - 5)",", "for" 3 ≤ x ≤ 6"," x ≠ 5),(= 10",", "for" x = 5),(=(x^2 - 3x - 10)/(x - 5)",", "for" 6 < x ≤ 9):}`
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Solution
`(x^2 - 3x - 10)/(x - 5)` is not defined at x = 5
∴ f(x) = `(x^2 - 3x - 10)/(x - 5)` where x ∈ [3, 5) ∪ (5, 6]
We can write f(x) explicitly, as follows:
f(x) `{:(=(x^2 - 3x - 10)/(x - 5),"," 3 ≤ x < 5),(= 10, "," x = 5),(= (x^2 - 3x - 10)/(x - 5), "," 5 < x ≤ 6),(= (x^2 - 3x - 10)/(x - 5), "," 6 < x ≤ 9):}`
∵ x2 – 3x – 10 = (x – 5) (x + 2)
∴ f(x) `{:(= x + 2",", 3 < x < 5),(= 10",", x = 5),(= x + 2",", 5 < x):}`
f(5) = 10
`lim_(x -> 5^-) "f"(x) = lim_(x -> 5^-) (x + 2)` = 5 + 2 = 7
`lim_(x -> 5^+) "f"(x) = lim_(x -> 5^+) (x + 2)` = 5 + 2 = 7
∴ f(5) = `lim_(x -> 5) "f"(x)`
∴ f(x) is continuous on its domain except at x = 5
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