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Evaluate the limit: lim(x->0)(sqrt(2)-sq...

Evaluate the limit: `lim_(x->0)(sqrt(2)-sqrt(1+cosx))/(sin^2x)`

Text Solution

Verified by Experts

The correct Answer is:
`1//4sqrt(2)`
We have `underset(xto0)lim(sqrt(2)-sqrt(1+cosx))/(sin^(2)x)`
`underset(xto0)lim(2-(1+cosx))/(sin^(2)x)xx(1)/(sqrt(2)+sqrt(1+cosx))`
`=underset(xto0)lim(1-cosx)/((1+cosx)(1-cosx))xxunderset(xto0)lim(1)/(sqrt(2)+sqrt(1+cosx))`
`=underset(xto0)lim(1)/((1+cosx))xx(1)/(2sqrt(2))`
`=(1)/(4sqrt(2))`
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