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Question
Let f(x) = ax + b (where a and b are unknown)
= x2 + 5 for x ∈ R
Find the values of a and b, so that f(x) is continuous at x = 1
Solution
f(x) `{:(= "a"x + "b", x < 1), (= x^2 + 5, x ≥ 1):}`
f(x) = x2 + 5
∴ f(x) = ax + b
Where, a = 1, b = 5
∴ f(1) = 1 + 5 = 6
L.H.L. = `lim_(x -> 1^-) "f"(x) = lim_(x -> 1^-) ("a"x + "b")` = a + b
R.H.L. = `lim_(x -> 1^+) "f"(x) = lim_(x -> 1^+) (x^2 + 5)` = 1 + 5 = 6
given, f(x) is continuous at n = 1
∴ L.H.L. = R.H.L.
∴ a + b = 6 where, a, b ∈ R
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