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Question
Evaluate the following limits:
`lim_(x -> 0) (tan x - sin x)/x^3`
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Solution
We know `lim_(x -> 0) sinx/x` = 1
`lim_(x -> 0) (tan x - sin x)/x^3 = lim_(x -> 0) (sinx/cosx - sin x)/x^3`
= `lim_(x -> 0) ((sinx - sinx cosx)/cosx)/x^3`
= `lim_(x -> 0) (sinx(1 - cosx))/(x^3 cosx)`
= `lim_(x -> 0) sinx/x * (2sin^2 (x/2))/(x^2) xx 1/cosx`
= `lim_(x -> 0) sinx/x xx (2sin^2 (x/2))/(2^2 xx x^2/2^2) xx 1/cosx`
= `lim_(x -> 0) sinx/x xx 1/2 (lim_(x/2 -> 0) (sin(x/2))/(x/2))^2 xx lim_(x - 0) 1/cosx`
= `1 xx 1/2 xx 1^2 xx 1/cos0`
= `1/2 xx 1/1`
`lim_(x -> 0) (tan x - sin x)/x^3 = 1/2`
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