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Evaluate lim(xto0)(sin^(-1)x-tan^(-1)x)/...

Evaluate `lim_(xto0)(sin^(-1)x-tan^(-1)x)/(x^(3)).`

Text Solution

Verified by Experts

`underset(xto0)lim(sin^(-1)x-tan^(-1)x)/(x^(3))" "=underset(xto0)lim((1)/(sqrt(1-x^(2)))-(1)/(1+x^(2)))/(3xx2)`
`=underset(xto0)lim((1+x^(2))-sqrt(1-x^(2)))/(3x^(2)sqrt(1-x^(2))(1+x^(2)))`
(Using L'Hospital's rule)
`=underset(xto0)lim((1+x^(2))-(1-x^(2)))/(3x^(2)sqrt(1-x^(2))(1+x^(2)))xx(1)/((1+x^(2))+sqrt(1-x^(2)))`
(Rstionalizing)
`=underset(xto0 )lim(x^(4)+3x^(2))/(3x^(2)sqrt(1-x^(2))(1+x^(2)))xx(1)/((1+x^(2))+sqrt(1-x^(2)))`
`=underset(xto0)lim(x^(2)+3)/(3x^(2)sqrt(1-x^(2))(1+x^(2)))xx(1)/((1+x^(2))+sqrt(1-x^(2)))=1//2`
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