Computing the limit or showing that the limit does not exist: $\lim_{(x,y)\to(0,0)}\frac{ x ^ 2 + 3 x - 4 y }{ x - y}$

Question

$$\lim_{(x,y)\to(0,0)}\frac{ x ^ 2 + 3 x - 4 y }{ x - y}$$

My try:

Since the limit does not exist when we substitute 0,0 I thought of proving that limit does not exists.

$y=0,x \to 0^{+}$

$f(x,0)=x+3$

$x=0,y \to 0^+$

$f(0,y)=4$

Since $f(x,0)\neq f(0,y)$ then limit does not exist.

Is my attempt correct?

Thanks..

Answer 1

You mean $$\lim_{x \to 0} f(x,0) = 3 \ne 4 = \lim_{y \to 0} f(0,y).$$

Since they are not equal when we travel along two trajectories, the limit doesn't exist.

Answer 2

With $m:=\dfrac yx$, you have

$$\frac{ x ^ 2 + 3 x - 4 y }{ x - y}=\frac{ x + 3 - 4 m }{ 1 - m}\to\frac{3-4m}{1-m}$$

which is not a constant function of $m$.