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

Answer 1

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