Evaluate `lim_(xtoa) (sinx-sina)/(sqrt(x)-sqrt(a))`

Explanation

Given, `underset(xtoa)"lim"(sinx-sina)/(sqrt(x)-sqrt(a)) =underset(xto0)"lim"((2cos(x+a))/2sin(x-a)/2)/(sqrt(x)-sqrt(a))`
`[therefore sinC-sinD=2cos(C+D)/2.sin(C-D)/2]`
`=underset(xto0)"lim"((2cos(x+a))/2sin(x-a)/2(sqrt(x)+sqrt(a)))/((sqrt(x)-sqrt(a))(sqrt(x)+sqrt(a)))`
`=underset(xto0)"lim"((2cos(x+a))/2sin(x-a)/2 (sqrt(x)+sqrt(a)))/(x-a)`
`=2underset(xto0)"lim"cos(x+a)/(2)(sqrt(x)+sqrt(a)).1/2underset(xto0)"lim"(sin((x-2)/2))/(2((x-a)/2))`
`=2underset(xto0)"lim"cos((x+a)/2)(sqrt(x)+sqrt(a)). 1/2underset(xto0)"lim"(sin((x-a)/2))/((x-a)/2)`
`=2.cosa/2.sqrt(a).1/2` `[therefore underset(xto0)"lim"(sinx)/x=1]`
`=sqrt(a)cosa/2`