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