Evaluate, `lim_(xto0) (sqrt(2)-sqrt(1+cosx))/(sin^(2)x)`

Explanation

Given, `underset(xto0)"lim"(sqrt(2)-sqrt(1+cosx))/(sin^(2)x)=underset(xto0)"lim"(sqrt(2)-sqrt(1+2cos^(2)x/2-1))/(sin^(2)x)` `[therefore cosx=2cos^(2)x/2-1]`
`=underset(xto0)"lim"(sqrt(2)=sqrt(2cos^(2)x/2))/(sin^(2)x)` `[therefore sinx=2sinx/2cosx/2]`
`=underset(xto0)"lim"(sqrt(2)(1-cosx/2))/(sin^(2)x)=underset(xto0)"lim"(sqrt(2)(1-1+2sin^(2)x/4))/(sin^(2)x)`
`=underset(xto0)"lim"(sqrt(2)(2sin^(2)x/4))/(sin^(2)x)=underset(xto0)"lim"2sqrt(2)(sin^(2)x/4)/(x/4)^(2).(x/4)^(2)/(sin^(2)x)`
`=2sqrt(2)underset(xto0)"lim"((sinx/4)/(x/4))^(2).underset(xto0)"lim"(x/(sinx))^(2).1/16`
`=2sqrt(2).1.1.1/16=1/(4sqrt(2))`