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Question
Evaluate the following limits: if `lim_(x -> 1)[(x^4 - 1)/(x - 1)] = lim_(x -> "a") [(x^3 - "a"^3)/(x - "a")]`, find all the value of a.
Solution
`lim_(x -> 1)[(x^4 - 1)/(x - 1)] = lim_(x -> "a") [(x^3 - "a"^3)/(x - "a")]`
`lim_(x -> 1) (x^4 - (1)^4)/(x - 1) = lim_(x -> "a") (x^3 - "a"^3)/(x - "a")`
∴ `4(1)^3 = 3"a"^2 ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
∴ 3a2 = 4
∴ a2 = `4/3`
∴ a = `± 2/sqrt(3)`
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