Question

The function `f(x) = (x^2 - 1)/(x^3 - 1)` is not defined at x = 1. What value must we give f(1) inorder to make f(x) continuous at x =1?

Answer 1

`f(x) = (x^2 - 1)/(x^3 - 1)`

f(x) is not defined at x = 1

`lim_(x -> 1) f(x) =  lim_(x -> 1) (x^2 - 1)/(x^3 - 1)`

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

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

= `(1 + 1)/(1^2 + 1 + 1)`

= `2/3`

`lim_(x -> 1) f(x) = 2/3`

The function f(x) has a removable discontinuity at x = 1.

Redefine f(x) as

`f(x) = {{:((x^2 - 1)/(x^3 - 1)",",  "if"  x ≠ 1),(2/3",",  "if"  x = 1):}`

∴ f(1) = `2/3`.

Then f(x) will be continuous at x = 1