# How can I prove this limit doesn't exist?

Right now, I’m doing a question:

$$\lim_{(x,y)\to(1,0)}\frac{xy-y}{(x-1)^2 +y^2}$$

I know the limit doesn’t exist, but I can’t figure out how to prove it. I tried putting $x=1$, and getting $0/y^2$, and put $y=0$, got $0/(x^2-2x+1)$, but I don’t think that does it.

(edit: this is not a duplicate; I’m having a hard time getting good explanations, hence why I asked)

#### Solutions Collecting From Web of "How can I prove this limit doesn't exist?"

Along the path $y=x-1$, we have

\begin{align} \lim_{(x,y)\to (1,0)}\frac{xy-y}{(x-1)^2+y^2}&=\lim_{(x,y)\to (1,0)}\frac{(x-1)^2}{2(x-1)^2}\\\\ &=\frac12 \end{align}

Along the path $y=0$, we have

$$\lim_{(x,y)\to (1,0)}\frac{xy-y}{(x-1)^2+y^2}=0$$

Take la sequence of points $P_n=(1\pm \frac 1n,\frac 1n)$. You have $$\lim_{x\to \infty}\frac{(1\pm \frac 1n)\frac 1n-\frac 1n}{(\pm\frac 1n)^2+((\frac 1n)^2}=\pm \frac 12$$ These two distinct limits say that the limit doesn’t exist.