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

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The limit `lim_(x->0) (sqrt 2 - sqrt(1 + cos x))/(sin^2x)` has to be determined.

`lim_(x->0) (sqrt 2 - sqrt(1 + cos x))/(sin^2x)`

substituting x = 0 gives the indeterminate form `0/0` . This allows the use of l'Hopital's rule and the denominator and numerator are substituted by their derivatives.

=> `lim_(x->0) (-1/2)*(-sin x)*(1/sqrt(1 + cos x))/(2*sin x*cos x)`

=> `lim_(x->0) (1/2)*(1/sqrt(1 + cos x))/(2*cos x)`

substituting x = 0

=> `(1/2)*(1/sqrt 2)/2`

=> `1/(4*sqrt 2)`

**The limit `lim_(x->0) (sqrt 2 - sqrt(1 + cos x))/(sin^2x)` = **`1/(4*sqrt 2)`

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