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Question
Evaluate the following limits:
`lim_(x -> ) (sinx(1 - cosx))/x^3`
Solution
We know `lim_(x -> 0) sinx/x` = 1
`lim_(x -> 0) (sinx(1 - cosx))/x^3 = lim_(x -> 0) (sinx xx 2 sin^2 x/2)/x^3`
= `lim_(x -> 0) (sinx/x) xx 2 (sin^2 x/2)/x^2`
= `lim_(x -> 0) (sinx/x) xx 2 (sin^2 x/2)/(2^2 xx x^2/2^2)`
= `lim_(x -> 0) [(sinx/x) xx 1/2 ((sin (x/2))/((x/2)))^2]`
= `lim_(x -> 0) (sinx/x) xx 1/2 (lim_(x/2 -> 0) (sin x/2)/(x/2))^2`
= `1 xx 1/2 xx 1`
`lim_(x -> ) (sinx(1 - cosx))/x^3 = 1/2`
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