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