Evaluate the following limits:
`lim_(xrarr pi)(sqrt(2+cosx)-1)/((pi-x)^(2))`

Answer

`1/4`

Explanation

Given limit `=lim_(xto(pi)/(4))((sqrt(2+cosx)-1)(sqrt(2+cosx)+1))/((pi-x)^(2)(sqrt(2+cosx)+1))`
`=lim_(xto pi)((1+cosx))/((pi-x)^(2)(sqrt(2+cosx)+1))=lim_(hto0)(1+cos(pi-h))/(h^(2){sqrt(2+cos(pi-h))+1})" "["putting"pi -x=h]`
`=lim_(hto0)((1-cosh))/{h^(2){sqrt(2-cosh)+1})=lim_(hto0)(2sin^(2)(h//2))/(((h)/(2))^(2)xx4xx{sqrt(2-cosh)+1})`
`=1/2lim_(hto0){(sin(h//2))/((h//2))}^(2).(1)/(lim_(hto0){sqrt(2-cosh)+1})=((1)/(2)xx1^(2)xx(1)/(2))=1/4.`