`lim_(y to 0) ((x+y)sec(x+y)-xsecx)/(y)`

Explanation

Given, `underset(yto0)"lim"((x+y)sec(x+y)-xsecx)/(y)`
`=underset(yto0)"lim"((x+y)/(cos(x+y))-x/(cosx))/(y)`
`=underset(yto0)"lim"[(xcosx+ycosx(x+y)+ycosx)/(ycosxcos(x+y))]`
`=underset(yto0)"lim"[(xcosx-xcos(x+y)+ycosx)/(ycosxcos(x+y)]]`
`=underset(yto0)"lim"(x[-2sin(x+y/2)sin(-y/2)]+ycosx)/(ycosxcos(x+y))` `[therefore cosC-cosD=-2sin(C+D)/(2).sin(C-D)/(2)]`
`=underset(yto0)"lim"[[(x{2sin(x+y/2)siny/2}+ycosx))/(ycosxcos(x+y)]]`
`=underset(yto0)"lim"(2xsin(x+y/2))/(cosxcos(x+y)).underset(yto0)"lim"(siny/2)/(y/2).1/2+underset(hto0)"lim"cos(x+h)`
`therefore underset(hto0)"lim"(sinx)/(x)=1` and `xto0 rArr kx to0]`
`=underset(yto0)"lim"(2xsin(x+y/2))/(cosxcos(x+y)).1/2+underset(yto0)"lim"sec(x+y)`
`(2xsinx)/(cosxcosx).1/2+secx`
`xtanxsecx+secx`
`=secx(xtanx+1)`