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प्रश्न
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?
उत्तर
`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
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