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Question
Find a and b if the following function is continuous at the point indicated against them.
`f(x) = (x^2 - 9)/(x - 3) + "a"` , for x > 3
= 5 , x = 3
= 2x2 + 3x + b , for x < 3
is continuous at x = 3
Solution
f is continuous at x = 3
∴ `f(3) = lim_(x→3^-) "f"(x)`
= `lim_(x→3^-) (2x^2 + 3x + "b")`
∴ 5 = 2(3)2 + 3(3) + b
∴ 5 = 18 + 9 + b
∴b = – 22
Also, f(3) = `lim_(x → 3^+) "f"(x)`
∴ 5 = `lim_(x → 3^+) (x^2 - 9)/(x - 3) + "a"`
= `lim_(x → 3^+) ((x + 3)(x - 3))/((x - 3)) + "a"`
= `lim_(x → 3^+) (x + 3) + "a" ...[(because x → 3";" x ≠ 3),(therefore x - 3 ≠ 0)]`
= (3 + 3) + a
∴ 5 = 6 + a
∴ a = – 1
∴ a = – 1, b = – 22
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