Question

At the given point x₀ discover whether the given function is continuous or discontinuous citing the reasons for your answer:

x₀ = 3, `f(x) = {{:((x^2 - 9)/(x - 3)",", "if"  x ≠ 3),(5",", "if"  x = 3):}`

Answer 1

`lim_(x -> 3^-) f(x) =  lim_(x -> 3^-) (x^2 - 9)/(x - 3)`

= `lim_(x -> 3^-) ((x + 3)(x - 3))/(x - 3)`

= `lim_(x -> 3^-) (x + 3)`

= 3 + 3

 = 6 .........(1)

`lim_(x -> 3^+) f(x) =  lim_(x -> 3^+) (x^2 - 9)/(x - 3)`

= `lim_(x -> 3^+) ((x + 3)(x - 3))/(x - 3)`

= `lim_(x -> 3^+) (x + 3)`

= 3 + 3

 = 6 .........(2)

From equations (1) and (2)

`lim_(x -> 3^-) f(x) =  lim_(x -> 3^-) f(x)` = 6

∴ `lim_(x -> 3) f(x)` = 6  ........(3)

`f(3)` = 5  ........(4)

From equations (3) and (4) we have

`lim_(x -> 3) f(x) ≠ f(3)`

∴ f(x) is not continuous at x₀ = 3.