Question:
Lt. y >0 (x +y) sec(x +y) + xsecx /y
Answer:
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
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