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Question
Evaluate: `lim_(x -> a) (sin x - sin a)/(sqrt(x) - sqrt(a))`
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Solution
Given that `lim_(x -> a) (sin x - sin a)/(sqrt(x) - sqrt(a))`
= `lim_(x -> a) (sin x - sin a)/(sqrt(x) - sqrt(a)) xx (sqrt(x) + sqrt(a))/(sqrt(x) + sqrt(a))`
= `lim_(x -> a) ((sin x - sin a)(sqrt(x) + sqrt(a)))/(x - a)`
= `lim_(x -> a) ((2 cos (x + a)/2 * sin (x - a)/2)(sqrt(x) + sqrt(a)))/(x - a)`
= `lim_((x -> a),(because (x - a)/2 -> 0)) (2 cos (x + a)/2 * (sin (x - a)/2)/(2 xx (x - a)/2)) (sqrt(x) + sqrt(a))`
= `lim_(x -> a) cos((x + a)/2)(sqrt(x) + sqrt(1))` .....`[because lim_((x - a)/2 -> 0) (sin (x - a)/2)/((x - a)/2) = 1]`
Taking limit we have
= `cos ((a + a)/2)(sqrt(a) + sqrt(a))`
= `cos a xx 2sqrt(a)`
= `2sqrt(a) * cos a`
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