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Question
`lim_(x -> 1) (x^m - 1)/(x^n - 1)` is ______.
Options
1
`m/n`
`- m/n`
`m^2/n^2`
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Solution
`lim_(x -> 1) (x^m - 1)/(x^n - 1)` is `m/n`.
Explanation:
Given, `lim_(x -> 1) (x^m - 1)/(x^n - 1)`
= `lim_(x -> 1) ((x^m - (1)^m)/(x - 1))/((x^n - (1)^n)/(x - 1))`
= `(m(1)^(m - 1))/(n(1)^(n - 1))`
= `m/n` .....`[because lim_(x -> a) (x^n - a^n)/(x - a) = na^(n - 1)]`
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