Determine the limits if they exist: lim_{(x,y)rightarrow(2,4)}frac{(x-2)^2(y-4)^2}{(x-2)^3+(y-4)^3}

hexacordoK 2020-10-19 Answered
Determine the limits if they exist: lim(x,y)(2,4)(x2)2(y4)2(x2)3+(y4)3
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Expert Answer

Ezra Herbert
Answered 2020-10-20 Author has 99 answers

Given that lim(x,y)(2,4)(x2)2(y4)2(x2)3+(y4)3
Take
(x2)=X and 4=Y then
when x2 then X0
when y4, then Y0
So, the equation (1) becomes
lim(x,y)>(2,4)(x2)2(y4)2(x2)3+(y4)3=lim(X,Y)(0,0)x2y2X3Y3
To check is limit path dependent or not, so put Y=mX
lim(X,Y)>(0,0)X2Y2X3+Y3=limX0X2m2X2X3+(mX)3=limX>0m2X4X3(1+m3)
lim(X,Y)>(0,0)X2Y2X3+Y3=limX0m2X4X3(1+m3)=0
lim(x,y)>(2,4)(x2)2(y4)2(x2)3+(y4)3=lim(X,Y)>(0,0)x2y2X3Y3=0
Hence, limit exist and
lim(x,y)(2,4)(x2)2(y4)2(x2)3+(y4)3=0

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