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Question Number 152407 by imjagoll last updated on 28/Aug/21

Answered by Olaf_Thorendsen last updated on 28/Aug/21

$$f(x,y)=\frac{(x+y)\sec (x+y)-x\sec x}{y}$$$$f(x,y)=\frac{\frac{x+y}{\cos (x+y)}-x\sec x}{y}$$$$f(x,y)=\frac{\frac{x+y}{\cos x\cos y-\sin x\sin y}-x\sec x}{y}$$$$f(x,y)\underset{y\to 0}{\sim }\frac{\frac{x+y}{\cos x-y\sin x}-x\sec x}{y}$$$$f(x,y)\underset{y\to 0}{\sim }\frac{\frac{x+y}{1-y\tan x}-x}{y\cos x}$$$$f(x,y)\underset{y\to 0}{\sim }\frac{(x+y)(1+y\tan x)-x}{y\cos x}$$$$f(x,y)\underset{y\to 0}{\sim }\frac{x+y(1+x\tan x)-x}{y\cos x}$$$$f(x,y)\underset{y\to 0}{\sim }\frac{1+x\tan x}{\cos x}$$

Answered by EDWIN88 last updated on 28/Aug/21

$$lety=h$$$$\underset{h\to 0}{\lim }\frac{(x+h)\sec (x+h)-x\sec x}{h}$$$$=\frac{d}{dx}\left(x\sec x\right)=\sec x+x\sec x\tan x$$$$=(1+x\tan x)\sec x$$

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