Question

For what value of `alpha` is this function `f(x) = {{:((x^4 - 1)/(x - 1)",",  "if"  x ≠ 1),(alpha",",  "if"  x = 1):}` continuous at x = 1?

Answer 1

`f(x) = {{:((x^4 - 1)/(x - 1)",",  "if"  x ≠ 1),(alpha",",  "if"  x = 1):}` 

Given that f(x) is continuous at x = 1.

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

`lim_(x -> 1^-) f(x)  =  lim_(x -> 1^-) (x^4 - 1)/(x - 1)`

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

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

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

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

`lim_(x -> 1^-) f(x)` = 2 × 2 = 4  .......(2)

`lim_(x -> 1^+) f(x) =  lim_(x -> 1^+) (x^4 - 1)/(x - 1)`

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

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

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

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

`lim_(x -> 1^-) f(x)` = 2 × 2 = 4  .......(3)

From equations (2) and (3) we have

`lim_(x -> 1^-) f(x) =  lim_(x -> 1^-) f(x)` = 4

∴ `lim_(x -> 1) f(x)` = 4

`f(1) = alpha`

∴ Equation (1) ⇒ 4 = `alpha`