In my vector calculus class my instructor informally claimed that given $2$ coprime multivariate polynomials $P(\mathbf x),Q(\mathbf x)$ with $P(\mathbf x_0)=Q(\mathbf x_0)=0$ and $\text{deg}(P)\le\text{deg}(Q)$,
$$\lim_{\mathbf x \to\mathbf x_0}\frac{P(\mathbf x)}{Q(\mathbf x)} \quad \text{does not exist.}$$
I tried to prove this by noting that we can assume $\mathbf x_0=\mathbf 0$ WLOG and then go along the path $x_i=kx_1$ for $i\ne1$. The limit becomes
$$\lim_{x_1 \to 0}\frac{P_1(x_1)}{Q_1(x_1)}$$ where $\text{deg}(P_1)\le \text{deg}(Q_1)$. However, this does not help. Can someone either provide a proof or show that the claim is false?
Here's a counterexample for when the degrees are equal: $$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{x^2+y^4}=0$$ and here's a counterexample for $\text{deg}(P)<\text{deg}(Q)$: $$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{x^2+y^6}=0$$