# Limit of function of two variables

## Homework Statement

Fnd the limit, if it exists, or show that the limit does not exist.
lim $$_{(x,y)--> (0,0)} [x^2/(x^2 + y^2)]$$

## The Attempt at a Solution

If x = 0, then f(0,y) = 0. f(x,y) --> 0 when (x,y) --> (0,0) along the y-axis.

If y = 0, then f(x,0) = 1. f(x,y) --> 1 when (,y) --> (0,0) along the x-axis.

1st, am I doing this right? By simply plugging in y = 0 or x = 0, I can determine the limit?

2nd, why am I not using L'Hospital's Rule. If (x,y) --> (0,0) then won't the function obviously go to 0/0 which is not real?

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You are doing it right. An easier way to see what is going on to make the respective polar subsititutions and take the limit as r->0

yes, you are correct. Indeed, the limit does not exist, since when taking different paths, the limits are different.

HallsofIvy
Just to clarify: if you take the limit along two different paths and get different answers then the limit itself does not exist. If you take limits along many different paths and always get the same answer, that does not prove the limit exists because you can't try all possible paths. If you suspect the limit does exist, then the best thing to do is what end3r7 suggested: change to polar coordinates so that the distance from (0,0) is determined by the single variable r. If the limit, as r goes to 0, does not depend on $\theta$ the limit exists.