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Evaluate lim(xto-1^(+))(sqrt(pi)-sqrt(co...

Evaluate `lim_(xto-1^(+))(sqrt(pi)-sqrt(cos^(-1)x))/(sqrt(1+x)).`

Text Solution

Verified by Experts

`underset(xto-1^(+))lim(sqrt(pi)-sqrt(cos^(-1)x))/(sqrt(1+x))`
`=underset(xto-1^(+))lim(pi-cos^(-1)x)/(sqrt(1+x)).(1)/(sqrt(pi)+sqrt(cos^(-1)x))`
`=underset(xto-1^(+))lim(cos^(-1)(-x))/(sqrt(1+x)).(1)/(sqrt(pi)+sqrt(pi))`
`=(1)/(2sqrt(pi))underset(thetato0^(+))lim(theta)/(sqrt(1-costheta))" "`(Putting `cos^(-1)(-x)=theta)`
`=(1)/(2sqrt(pi))underset(thetato0^(+))lim(theta)/(sqrt(2"sin"^(2)(theta)/(2)))`
`=(1)/(sqrt(2pi))underset(thetato0^(+))lim((theta)/(2))/("sin"(theta)/(2))`
`=(1)/(sqrt(2pi))`
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