Find the following using L'Hospital's rule
`(i) lim_(xto0) ((16+5x)^(1//4)-2)/((32+3x)^(1//5)-2)" "(ii)lim_(xto pi//2) [xtanx-(pi//2)secx]`

Explanation

`(i) underset(xto0)lim((16+5x)^(1//4)-2)/((32+3x)^(1//5)-2)" "`(0/0 form)
`(i) underset(xto0)lim((1)/(4)(16+5x)^((1)/(4)-1)xx5)/((1)/(5)(32+3x)^((1)/(5)-1)xx3)" "`(Using L'Hospital rule)
`=((1)/(4)(16)-(3)/(4)xx5)/((1)/(5)(32)-(4)/(5)xx3)=((1)/(4)xx(1)/(8)xx5)/((1)/(5)xx(1)/(16)xx3)=(25)/(6)`
`(ii)underset(xto pi//2)lim[xtanx-(pi//2)secx]`
`=underset(xto pi//2)lim[(xsinx)/(cosx)-(pi)/(2)xx(1)/(cosx)]`
`=underset(xto pi//2)lim(xsinx-pi//2)/(cosx)" "`(0/0 form)
`=underset(xto pi//2)lim(x.cosx+sinx.1-0)/(-sinx)" "`(Using L'Hospital rule)
`=(pi//2.cos(pi//2)+sinpi//2)/(-sin(pi//2))=(0+1)/(-1)=-1`