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)

Question

sergij07

4 days ago

12

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

Mathematics

2 answers:

tester [3.9K] · Answer 1 · 2026-03-01T01:16:38+03:00

It appears the limit to compute is

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

Taking an arbitrary line that passes through the origin $y=mx$ , we can rewrite the expression 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 limit's value is determined by the slope $m$ selected for the line, signifying that the limit is not independent of the path and therefore does not exist.

Svet_ta [4.3K] · Answer 2 · 2026-03-01T01:18:48+03:00

Answer:

As (x,y) approaches (0,0), the limit of f(x,y) is equal to 0.

Step-by-step explanation:

The function is defined as f(x, y) = x*y/(x^2 + 2*y^2).

The corresponding data is illustrated in the attached graphic.

When y is fixed, varying x causes f(x,y) to switch between positive and negative values, and the same holds true when x is fixed and y varies. This suggests that as (x,y) approaches (0,0), the limit can be conjectured to be 0.