Question

Show that the function have no limit as(x,y)->(0,0)
g(x,y)=x^2-y/x-y ​

Answer 1

Step-by-step explanation:

If f (x, y) =xy/ (x^2+y^2) when (x, y) not equal to zero and f(x,y) =0,when (x, y) =0 show that the f (x, y) is continuous at any point except the origin?

Note that f(x,y) is a rational function - the quotient of two polynomials in two variables. Polynomials are everywhere continuous functions, and quotients of continuous functions are continuous with the possible exception of where the denominator vanishes.

For f(x,y) , the only possible exception is where x2+y2=0 , or at (x,y)=(0,0) . So f(x,y) is continuous everywhere in R2∖{(0,0)} ; and we need to investigate continuity of f at (0,0) .

If f is to be continuous at (0,0) , we must have lim(x,y)→(0,0)f(x,y)=f(0,0)=0 . This would mean that, given any ϵ>0 , there is some neighbourhood of (0,0) - sets of the form {(x,y):x2+y2−−−−−−√<δ} , with δ>0 - such that

|f(x,y)|<ϵwheneverx2+y2−−−−−−√<δ

for a suitable choice of δ .

So basically you want f(x,y) arbitrarily small

Answer 2

Step-by-step explanation:

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