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Question
Evaluate the following limits:
`lim_(x -> 0) (1 - cos^2x)/(x sin2x)`
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Solution
We know `lim_(x -> 0) (sin x)/x` = 1
`lim_(x -> 0) (1 - cos^2x)/(x sin2x) = lim_(x -> 0) (sin^2x)/(x xx 2sinx cosx)`
= `1/2 lim_(x -> 0) (sin)/(xcosx)`
= `1/2 (lim_(x -> 0) (sinx)/x) (lim_(x -> 0) 1/cosx)`
= `1/2 xx 1 xx 1/cos0`
`lim_(x -> 0) (1 - cos^2x)/(x sin2x) = 1/2 xx 1/1`
= `1/2`
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