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Question
Evaluate: `lim_(x -> 0) (1 - cos mx)/(1 - cos nx)`
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Solution
Given that `lim_(x -> 0) (1 - cos mx)/(1 - cos nx)`
= `lim_(x -> 0) ((2 sin^2 m/2 x)/(2 sin^2 n/2 x))`
= `lim_(x -> 0) ((sin m/2 x)/(sin n/2 x))`
= `(lim_(x -> 0) ((sin m/2 x)/(m/2 x) xx m/2 x)^2)/(lim_(x -> 0) ((sin n/2 x)/(sin n/2 x) xx n/2 x)^2)`
= `(1 * m^2/4 x^2)/(1 * n^2/4 x^2)` ......`[because lim_(x -> 0) sinx/x = 1]`
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