A valid method of finding limits in two variables functions? I was wondering if in finding the limit of a two variables function (say, $F(x,y)$), I can choose the path by let $y=f(x)$, then find the limit in the same way of that in one variable functions.
For example,
$$
\lim_{(x,y) \to (0,0)} \frac{xy}{x^2+xy+y^2}
$$
(It has no limit there by choosing first $y=0$ and then $y=x$)
So I'm asking if the following procedures are correct:
Let $y=f(x)$ where $f(0)=0$ since the function passes $(0,0)$
The function then becomes:
$$
\frac{xf(x)}{x^2 + xf(x) +f(x)^2}
$$
Then it's an indeterminate form $[0/0]$, so I differentiate,
$$
\frac{xf'(x)+f(x)}{2x + xf'(x)+f(x) +2f(x)f'(x)}
$$
Then it's still $[0/0]$ so I differentiate again,
$$\frac{xf''(x)+2f'(x)}{2+xf''(x)+2f'(x)
+2f(x)f"(x)+2f'(x)^2}$$
By substituting $x=0$, I get
$$\lim F(x,y) = \frac{2f'(x)}{2+2f'(x)+2f'(x)^2}$$
Since $f'(x)$ depends on the path I choose, the limit depends on the path I choose also.
Thus, the limit at $(0,0)$ does not exist.
So that's all, the question I have are


*

*is this a valid method to determine existence of limits?

*is this a valid method to find the limit?


(My teacher says it wont work for 2.) but I'm still unclear about his explanations)
(Sorry if I made any mistake or this is a very stupid question, I'm very new to this site and this is my first question, thank you in advance!)

If the limit does not exist, you can find two paths that disagree. However, in case the limit does exit, your failure to find such paths is not a proof of anything. You will need to show that the limits are the same for all paths.
