Evaluate `lim_(xto(pi/3)) sqrt(1-cos6x)/(sqrt(2)(pi/3-x)`

Explanation

Given, `underset(xto(pi/3))"lim"sqrt(1-cos6x)/(sqrt(2)(pi/3-x)) = underset(xto3)"lim"sqrt(1-1+2sin^(2)+3x)/(sqrt(2)(pi/3-x))` `[therefore cos2x=1-2sin^(2)x]`
`=underset(xto(pi)//3)"lim"(sqrt(2)sin3x)/(sqrt(2)(pi/3-x)=underset(xto(pi)/3)"lim"(sin3x)/(pi/3-x)` `[therefore cos2x=1-2sin^(2)x]`
`=underset(xto(pi)//3)"lim"(sqrt(2)sin3x)/(sqrt(2)(pi/3)-x) = underset(xto(pi)/3)"lim"(sin3x)/(pi/3-x)`
`=underset(xto(pi//3))"lim" (sin(pi-3x))/(pi-3x)=3 xx 1` `[therefore underset(xto0)"lim"(sinx)/(x) =1]`
=3 `[therefore xto(pi/3) rArr (x-pi/3)to 0]`