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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?
Solution
`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`
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