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Question
Evaluate: `lim_(x -> pi/3) (sqrt(1 - cos 6x))/(sqrt(2)(pi/3 - x))`
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Solution
Given that `lim_(x -> pi/3) (sqrt(1 - cos 6x))/(sqrt(2)(pi/3 - x))`
= `lim_(x -> pi/2) (sqrt(2 sin^2 3x))/(sqrt(2) (pi/3 - x))` ......`[because 1 - cos theta = 2 sin^2 theta/2]`
= `lim_(x -> pi/3) (sqrt(2) sin 3x)/(sqrt(2)((pi - 3x)/3))`
= `lim_((x -> pi/3),(because pi - 3x -> 0)) (3 * sin (pi - 3x))/(pi - 3x)`
= 3 .....`[because lim_(x -> 0) sinx/x = 1]`
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