`lim_(xto1)(1-sqrt(x))/((cos^(-1)sqrt(x))^2)` का मान ज्ञात कीजिए ।

Explanation

माना `f(x) = (1-sqrt(x))/((cos^(-1)sqrt(x))^2)`

`x = 1` रखने पर

`f(x) = 0/0`

अब यहाँ माना `cos^(-1)sqrt(x) = theta`

अतः `x=cos^2 theta to 1`

`lim_(xto1)(1-sqrt(x))/((cos^(-1)sqrt(x))^2)`

`=lim_(theta to 0) (1-cos theta)/(theta^2)`

`rArr cos^2theta to 1 rArr theta to 0`

`= lim_(xto 0)(2 sin^2.(theta)/(2))/(theta^2)" "[(0)/(0)]`

`=lim_(thetato0)2[(sin.(theta)/(2))/(theta)]^2`

`=lim_(thetato0)2[(sin.(theta)/(2))/(2.(theta)/(2))]^2`

`=lim_(thetato0)2/4 [(sin.(theta)/(2))/((theta)/(2))]^2`

`=(1)/(2)`