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

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

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..

• You can obtain $\to$ by typing $to$. – N. F. Taussig Sep 27 '18 at 10:28
• Though not written with perfect rigor, you are right. – Yves Daoust Sep 27 '18 at 10:35

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