Counterexamples for $\lim f(x,y)=\lim \left(y\cdot \frac{f(x,y)}{y}\right)$.

Question

What are some examples of functions $f(x,y)$ satisfying the conditions below?

1. $\displaystyle\lim_{(x,y)\to(0,0)} f(x,y)$ does not exist.
2. $\displaystyle\lim_{(x,y)\to(0,0)} \frac{f(x,y)}{y}$ exists.

Is there any example which is not defined by parts? In other words, is there any example which does not have the form below? $$f(x,y)=\left\{\begin{align*} g(x,y)\text{ if...}\\ h(x,y)\text{ if...} \end{align*}\right.$$

I want this example to show that, despite giving the correct answer, the solution \begin{align*} \lim_{(x,y)\to(0,0)} \frac{\sin(xy)}{x}&=\lim_{(x,y)\to(0,0)} y\cdot\frac{\sin(xy)}{xy}\\ &=\left(\lim_{(x,y)\to(0,0)} y\right)\cdot\left(\lim_{(x,y)\to(0,0)}\frac{\sin(xy)}{xy}\right)\\&=0\cdot\left(\lim_{u\to 0}\frac{\sin(u)}{u}\right)\\ &=0\cdot 1\\ &=0 \end{align*} is wrong (unless we know that the limit exists). So, what is the best example to illustrate that the manipulation $$\lim_{(x,y)\to(0,0)} f(x,y)=\lim_{(x,y)\to(0,0)} y\cdot \frac{f(x,y)}{y}$$ can fail?

Answer 1

Let, the domain of $f$ be $D$. Then the domain for the function $\frac{f(x,y)}{y}$ is $D\setminus\{(x,y)\in D:y=0\}$. Hence, the domain for the function $y.\frac{f(x,y)}{y}$ is $D\cap\big(D\setminus\{(x,y)\in D:y=0\}\big)=D\setminus\{(x,y)\in D:y=0\}$. On the other hand, the domain for $f(x,y)$ is $D$. So, when we say $\lim_{(x,y)\rightarrow (0,0)}f(x,y)$, we mean the limit inside the domain $D$, but for $\lim_{(x,y)\rightarrow (0,0)}y.\frac{f(x,y)}{y}$, we mean the limit inside the domain $D\setminus\{(x,y)\in D:y=0\}$. Hence, the two limit may differ. And they will be equal if we can continuously extend $y.\frac{f(x,y)}{y}$ to $f$ on $\big(D\setminus (0,0)\big)$(at least locally near 0).