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Question
Evaluate the following limits:
`lim_(x ->) (x^"m" - 1)/(x^"n" - 1)`, m and n are integers
Solution
`lim_(x ->) (x^"m" - 1)/(x^"n" - 1) = lim_(x -> 1) (x^"m" - 1^"m")/(x^"n" - 1^"n")`
= `lim_(x -> 1) (x^"m" - 1^"m")/(x - 1) xx (x - 1)/(x^"n" - 1^"n")`
= `lim_(x -> 1) ((x^"m" - 1^"m")/(x - 1)) xx 1/(lim_(x -> 1) (x^"n" - 1^"n")/(x - 1))`
`lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n"- 1)`
= `"m"(1)^("m" - 1) xx 1/("n"(1)^("n" - 1)`
`lim_(x ->) (x^"m" - 1)/(x^"n" - 1) = "m"/"n"`
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