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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) = (((x+y)sec(x+y)−xsecx)/y) f(x,y) = ((((x+y)/(cos(x+y)))−xsecx)/y) f(x,y) = ((((x+y)/(cosxcosy−sinxsiny))−xsecx)/y) f(x,y) ∼_(y→0) ((((x+y)/(cosx−ysinx))−xsecx)/y) f(x,y) ∼_(y→0) ((((x+y)/(1−ytanx))−x)/(ycosx)) f(x,y) ∼_(y→0) (((x+y)(1+ytanx)−x)/(ycosx)) f(x,y) ∼_(y→0) ((x+y(1+xtanx)−x)/(ycosx)) f(x,y) ∼_(y→0) ((1+xtanx)/(cosx))

$${f}\left({x},{y}\right)\:=\:\frac{\left({x}+{y}\right)\mathrm{sec}\left({x}+{y}\right)−{x}\mathrm{sec}{x}}{{y}} \\ $$$${f}\left({x},{y}\right)\:=\:\frac{\frac{{x}+{y}}{\mathrm{cos}\left({x}+{y}\right)}−{x}\mathrm{sec}{x}}{{y}} \\ $$$${f}\left({x},{y}\right)\:=\:\frac{\frac{{x}+{y}}{\mathrm{cos}{x}\mathrm{cos}{y}−\mathrm{sin}{x}\mathrm{sin}{y}}−{x}\mathrm{sec}{x}}{{y}} \\ $$$${f}\left({x},{y}\right)\:\underset{{y}\rightarrow\mathrm{0}} {\sim}\:\frac{\frac{{x}+{y}}{\mathrm{cos}{x}−{y}\mathrm{sin}{x}}−{x}\mathrm{sec}{x}}{{y}} \\ $$$${f}\left({x},{y}\right)\:\underset{{y}\rightarrow\mathrm{0}} {\sim}\:\frac{\frac{{x}+{y}}{\mathrm{1}−{y}\mathrm{tan}{x}}−{x}}{{y}\mathrm{cos}{x}} \\ $$$${f}\left({x},{y}\right)\:\underset{{y}\rightarrow\mathrm{0}} {\sim}\:\frac{\left({x}+{y}\right)\left(\mathrm{1}+{y}\mathrm{tan}{x}\right)−{x}}{{y}\mathrm{cos}{x}} \\ $$$${f}\left({x},{y}\right)\:\underset{{y}\rightarrow\mathrm{0}} {\sim}\:\frac{{x}+{y}\left(\mathrm{1}+{x}\mathrm{tan}{x}\right)−{x}}{{y}\mathrm{cos}{x}} \\ $$$${f}\left({x},{y}\right)\:\underset{{y}\rightarrow\mathrm{0}} {\sim}\:\frac{\mathrm{1}+{x}\mathrm{tan}{x}}{\mathrm{cos}{x}} \\ $$

Answered by EDWIN88 last updated on 28/Aug/21

let y=h lim_(h→0) (((x+h)sec (x+h)−x sec x)/h) = (d/dx)(x sec x)=sec x +x sec x tan x = (1+x tan x)sec x

$${let}\:{y}={h}\: \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left({x}+{h}\right)\mathrm{sec}\:\left({x}+{h}\right)−{x}\:\mathrm{sec}\:{x}}{{h}}\: \\ $$$$=\:\frac{{d}}{{dx}}\left({x}\:\mathrm{sec}\:{x}\right)=\mathrm{sec}\:{x}\:+{x}\:\mathrm{sec}\:{x}\:\mathrm{tan}\:{x} \\ $$$$\:=\:\left(\mathrm{1}+{x}\:\mathrm{tan}\:{x}\right)\mathrm{sec}\:{x} \\ $$

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