Question

Question #26742

Answer 1

`=(1+x tanx)/cos x`

Explanation:

`((x+y) sec(x+y)-xsecx)/y=`
`((x+y)cosx-x cos(x+y))/(y cosx cos(x+y))=`
`((x+y)cos x-x(cos x cosy-sin x sin y))/(y cosx cos(x+y))=`
`(xcos x(1-cosy)+y cosx + x sinx sin y)/(y cosx cos(x+y))=`
`=x/cos(x+y)((1-cosy)/y)+1/cos(x+y)+(x sinx)/(cos x cos(x+y)) siny/y`

but

`lim_(y->0)(1-cosy)/y=0` and
`lim_(y->0)(siny)/y=1`

so

`lim_(y->0)((x+y) sec(x+y)-xsecx)/y=(1+x tanx)/cos x`