lim(y->0)((x+y)sec(x+y)-xsecx)/y is equa...

`lim_(y->0)((x+y)sec(x+y)-xsecx)/y` is equal to (a) `secx(xtanx+1)` (b) `xtanx+secx` (c) `xsecx+tanx` (d) none of these

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The correct Answer is:

A

`underset(yto0)lim{(x{sec(x+y)-secx})/(y)+sec(x+y)}`
`=underset(yto0)lim[(x)/(y){(cosx-cos(x+y))/(cos(x+y)cosx)}]+underset(yto0)limsec(x+y)`
`=underset(yto0)lim[(x2sin(x+(y)/(2))sin((y)/(2)))/(ycos(x+y)cosx)]+secx`
`=underset(yto0)lim[(xsin(x+(y)/(2)))/(cos(x+y)cosx)xx(sin((y)/(2)))/((y)/(2))]+secx`
`=xtanxsecx+secx`
`=secx(xtanx+1)`

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