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Question
`lim_(x -> 0) (x^2 cosx)/(1 - cosx)` is ______.
Options
2
`3/2`
`(-3)/2`
1
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Solution
`lim_(x -> 0) (x^2 cosx)/(1 - cosx)` is 2.
Explanation:
Given `lim_(x -> 0) (x^2 cosx)/(1 - cosx)`
= `lim_(x -> 0) (x^2 cosx)/(2sin^2 x/2)` .....`[because 1 - cos x = 2 sin^2 x/2]`
= `lim_(x -> 0) (x^2/4 xx 4 cos x)/(2 sin^2 x/2)`
= `lim_(x -> 0 => x/2 -> 0) ((x/2)^2 * 2 cos x)/(sin^2 x/2)`
= `lim_(x/2 -> 0) ((x/2)/(sin x/2))^2 * 2 cos x`
= 2 cos 0
= `2 xx 1`
= 2 ......`[because lim_(x -> 0) x/sinx = 1]`
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