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Question
`lim_(x -> 0) (1 - cos 4theta)/(1 - cos 6theta)` is ______.
Options
`4/9`
`1/2`
`(-1)/2`
–1
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Solution
`lim_(x -> 0) (1 - cos 4theta)/(1 - cos 6theta)` is `4/9`.
Explanation:
Given `lim_(theta -> 0) (1 - cos 4theta)/(1 - cos 6theta)`
= `lim_(theta -> 0) (2sin^2 2theta)/(2sin^2 3theta)` .....`[because 1 - cos theta = 2 sin^2 theta/2]`
= `lim_(theta -> 0) (sin^2 2theta)/(sin^2 3theta)`
= `lim_(theta -> 0) [(sin 2 theta)/(sin 3theta)]^2`
= `lim_((theta -> 0),(2theta -> 0),(3theta -> 0)) [((sin 2theta)/(2theta) xx 2theta)/((sin 3theta)/(3theta) xx 3theta)]^2`
= `[(2theta)/(3theta)]^2`
= `(2/3)^2`
= `4/9`
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