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Question
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 + sinx) - sqrt(1 - sinx))/tanx`
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Solution
`lim_(x -> 0) (sqrt(1 + sinx) - sqrt(1 - sinx))/tanx = lim_(x -> 0) ((sqrt(1 + sinx) sqrt(1 - sinx))(sqrt(1 + sinx) + sqrt(1 - sinx)))/(tanx(sqrt(1 - sinx) + sqrt(1 - sinx))`
= `lim_(x -> 0) ((1 + sinx) - (1 -sinx))/(sinx/cosx (sqrt(1 + sinx) + sqrt(1 - sin))`
= `lim_(x -> 0) (cosx[1 + sinx - 1 + sinx])/(sinx(sqrt(1 + sinx) + sqrt(1 - sinx))`
= `lim_(x -> 0) (cosx xx 2sinx)/(sinx(sqrt(1 + sinx) + sqrt(1 - sinx))`
= `2 lim_(x -> 0) cosx/((sqrt(1 + sinx) + sqrt(1 - sinx))`
= `2 x (cos 0)/((sqrt(1 + sin0) + sqrt(1 - sin))`
= `(2 xx 1)/((sqrt(1 + 0) + sqrt(1 - 0))`
= `2/(1 +1)`
= `2/2`
`lim_(x -> 0) (sqrt(1 + sinx) - sqrt(1 - sinx))/tanx` = 1
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