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Question
Evaluate the following limits: if `lim_(x -> 5)[(x^"k" - 5^"k")/(x - 5)]` = 500, find all possible values of k.
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Solution
`lim_(x -> 5)[(x^"k" - 5^"k")/(x - 5)]` = 500
∴ `"k"(5)^("k" - 1) = 500 ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a")] = "na"^("n" - 1)]`
∴ k(5)k–1 = 4 x 125
∴ k(5)k–1 = 4 x (5)3
∴ k(5)k–1 = 4 x (5)4–1
Comparing both sides, we get
k = 4
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