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

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?

• Do you want your $g$ and $h$ to be continuous? Apr 7, 2019 at 0:26
• @Surajit In fact I wonder if there is some example which is not separated in two parts ($g$ and $h$). Apr 7, 2019 at 0:33

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).