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प्रश्न
Evaluate the following limit :
If `lim_(x -> 5) [(x^"k" - 5^"k")/(x - 5)]` = 500, find all possible values of k.
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उत्तर
`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 × 125
∴ k(5)k–1 = 4 × (5)3
∴ k(5)k–1 = 4 × (5)4–1
Comparing both sides, we get
k = 4
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