Question

lim (x+y) sec (x+y) -x sec x / y

y-->0

Answer 1

Given, limy->0 {(x + y) * sec (x + y) - x*sec x}/y

= limy->0 {x sec (x + y) + y sec (x + y) - x*sec x}/y 

= limy->0 [x{ sec (x + y) - sec x} + y sec (x + y)]/y

= limy->0 x{ sec (x + y) - sec x}/y + limy->0 {y sec (x + y)}/y

= limy->0 x{1/cos (x + y) - 1/cos x}/y + limy->0{y sec (x + y)}/y

= limy->0 [{cos x - cos (x + y)} * x/{y*cos (x + y)*cos x}] + limy->0 {y sec (x + y)}/y

= limy->0 [{2sin (x + y/2) * sin(y/2)} * 2x/{2y*cos (x + y)*cos x}] + limy->0 {y sec (x + y)}/y

= limy->0 {sin (x + y/2) * limy->0{sin(y/2)/(2y/2)} * limy->0 { x/{y*cos (x + y)*cos x}] + sec x

= sin x * 1 * x/cos2 x + sec x

= x*tan x*sec x + sec x

So, limy->0 {(x + y) * sec (x + y) - x*sec x}/y = x*tan x*sec x + sec x

Answer 2

Answer:

plz see the answer above