Question

Find the following limits: `(i)lim_(xto0) (1)/(x)sin^(-1)((2x)/(1+x^(2)))" "(ii)lim_(xto0) (1)/(x)sin^(-1)(3x-4x^(3))`

Answer 1

(i) We know that `sin^(-1)((2x)/(1+x^(2)))=2tan^(-1)x," for "-1lexle1`
`implies" "underset(xto0)lim(1)/(x)sin^(-1)((2x)/(1+x^(2)))=underset(xto0)lim(2tan^(-1)x)/(x)=2`
(ii) We know that `sin^(-1)(3x-4x^(3))=3sin^(-1)x," for "-1//2lexle1//2`
`implies" "underset(xto0)lim1/xsin^(-1)(3x-4x^(3))=underset(xto0)lim(3sin^(-1)x)/(x)=3`