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Question
Find the constant b that makes g continuous on `(- oo, oo)`.
`g(x) = {{:(x^2 - "b"^2,"if" x < 4),("b"x + 20, "if" x ≥ 4):}`
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Solution
`g(x) = {{:(x^2 - "b"^2,"if" x < 4),("b"x + 20, "if" x ≥ 4):}`
Given g is continuous on R.
∴ g(x) is continuous at x = 4.
`lim_(x -> 4^-) g(x) = lim_(x -> 4^+) g(x)`
`lim_(x ->4^-) (x^2 - "b"^2) = lim_(x -> 4^+) ("b"x + 20)`
42 – b2 = b × 4 + 20
16 – b2 = 4b + 20
b2 + 4b + 20 – 16 = 0
b2 + 4b + 4 = 0
(b + 2)2 = 0
b + 2 = 0 ⇒ b = – 2
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