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Question
`lim_(x -> 0) sinx/(sqrt(x + 1) - sqrt(1 - x)` is ______.
Options
2
0
1
–1
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Solution
`lim_(x -> 0) sinx/(sqrt(x + 1) - sqrt(1 - x)` is 1.
Explanation:
Given `lim_(x -> 0) sinx/(sqrt(x + 1) - sqrt(1 - x))`
= `lim_(x -> 0) (sinx [sqrt(x + 1) + sqrt(1 - x)])/((sqrt(x + 1) - sqrt(1 - x))(sqrt(x + 1) + sqrt(1 - x))`
= `lim_(x -> 0) (sin x[sqrt(x + 1) + sqrt(1 - x)])/(x + 1 - 1 + x)`
= `lim_(x -> 0) (sin x[sqrt(x + 1) + sqrt(1 - x)])/(2x)`
= `1/2 * lim_(x -> 0) sinx/x [sqrt(x + 1) + sqrt(1 - x)]`
Taking limit, we get
= `1/2 xx 1 xx [sqrt(0 + 1) + sqrt(1 - 0)]`
= `1/2 xx 1 xx 2`
= 1
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