Question

Find the following limits: `(i) lim_(xto0) (1-x)^((1)/(x))" "(ii) lim_(xto1) (1+log_(e)x)^((1)/(log_(e)x))`
`(iii)lim_(xto0) (1+sinx)^((1)/(x))`

Answer 1

`(i) underset(xto)lim(1-x)^((1)/(x))=(underset(xto0)lim(1-x)^((1)/(-x)))^(-1)=e^(-1)`
(ii) Since, `underset(xto1)log_(e)x=0, underset(xto1)lim(1+log_(e)x)^((1)/(log_(e)x))=1`
`(iii)underset(xto0)lim (1+sinx)^((1)/(x))=underset(xto0)lim((1+sinx)^((1)/(sinx)))^((sinx)/(x))`
`=(underset(xto0)lim(1+sinx)^((1)/(sinx)))^(underset(xto0)lim^((sinx)/(x)))`
`=e^(1)=e`