Question

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

x₀ = 1, `f(x) = {{:((x^2 - 1)/(x - 1)",", x ≠ 1),(2",", x = 1):}`

Answer 1

`lim_(x -> 1^-) f(x) =  lim_(x -> 1^-) (x^2 - 1)/(x - 1)`

= `lim_(x -> 1^-) ((x - 1)(x + 1))/((x - 1))`

= `lim_(x -> 1^-) (x + 1)`

= 1 + 1

= 2

`lim_(x -> 1^+) f(x) =  lim_(x -> 1^+) (x^2 - 1)/(x - 1)`

= `lim_(x -> 1^+) ((x - 1)(x + 1))/((x - 1))`

= `lim_(x -> 1^+) (x + 1)`

= 1 + 1

= 2

`lim_(x -> 1^-) f(x) =  lim_(x -> 1^+) f(x)` = 2

Hence `lim_(x -> 1) f(x)` = 2   ........(1)

`f(1)` = 2  ........(2)

From equation (1) and (2)

`lim_(x -> 1) f(x) = f(1)`

∴ f(x) is continuous at x₀ = 1