Question

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

Answer 1

Seems to be that the limit to compute is

$\displaystyle\lim_{(x,y)\to(0,0)}\frac{xy}{x^2+2y^2}$

Consider an arbitrary line through the origin $y=mx$, so that we rewrite the above as

$\displaystyle\lim_{x\to0}\frac{mx^2}{x^2+2m^2x^2}=\lim_{x\to0}\frac m{1+2m^2}=\frac m{1+2m^2}$

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

Answer 2

Answer:

The limit of f(x,y) as (x,y) → (0,0) = 0

Step-by-step explanation:

f(x, y) = x*y/(x^2 + 2*y^2)

The table can be seen in the figure attached.

If y = constant, changes in x values makes f(x,y) changes from positive values to negative values or vice versa. If x = constant, changes in y values makes the same effect. From that information, it seems that lim (x,y)->(0,0) = 0.