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Evaluate lim(xto0^(+)) (1)/(x)cos^(-1)((...

Evaluate `lim_(xto0^(+)) (1)/(x)cos^(-1)((sinx)/(x))`.

Text Solution

Verified by Experts

`underset(xto0^(+))lim(1)/(x)cos^(-1)((sinx)/(x))`
`=underset(xto0^(+))lim(sin^(-1)(sqrt(1-(sin^(2)x)/(x^(2)))))/(x)`
`=underset(xto0^(+))lim(sin^(-1)(sqrt(1-(sin^(2)x)/(x^(2))))sqrt(1-(sin^(2)x)/(x^(2))))/sqrt(1-(sin^(2)x)/(x^(2)))`
`=underset(xto0^(+))lim(sqrt(x^(2)-sin^(2)x))/(x^(2))`
`=underset(xto0^(+))limsqrt((x+sinx)/(x).(x-sinx)/(x^(3)))`
`=sqrt(2)underset(xto0^(+))limsqrt((x-(x-(x^(3))/(3!)))/(x^(3)))`
`=sqrt(2)sqrt((1)/(6))`
`=sqrt((1)/(3))`
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