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Use a table of numerical values of f(x,y) for (x,y) near the origin to make a conjecture about the value of the limit of f(x,y) as (x,y) → (0,0) . (if the limit does not seem to exist, enter dne.) f(x, y) = x y x2 + 2 y2

Respuesta :

LammettHash
Seems to be that the limit to compute is

[tex]\displaystyle\lim_{(x,y)\to(0,0)}\frac{xy}{x^2+2y^2}[/tex]

Consider an arbitrary line through the origin [tex]y=mx[/tex], so that we rewrite the above as

[tex]\displaystyle\lim_{x\to0}\frac{mx^2}{x^2+2m^2x^2}=\lim_{x\to0}\frac m{1+2m^2}=\frac m{1+2m^2}[/tex]

The value of the limit then depends on the slope [tex]m[/tex] of the line chosen, which means the limit is path-dependent and thus does not exist.

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