Find $$\lim\limits_{z\to0}\frac{\mathrm{Re}(z)\cdot\mathrm{Im}(z)}{\mathrm{Re}(z)+\mathrm{Im}(z)}.$$
Let $z= x + i\cdot y$, then $$\lim_{x,y \to 0} \frac{x\cdot y}{x+y}.$$
Taking along $y=m\cdot x$, then $$\lim_{x \to 0} \frac{m\cdot x}{1+m}.$$ This is clearly equal to $0$ , i.e. independent of $m$ . But what happens when $m= -1$?
Even interpreting $\lim$ as "limit in the subset $x + y\ne 0$", the quotient $xy/(x + y)$ will take values arbitrarily big. Namely, for example, when $y = x^4 - x$ , $x\ne 0$: $$\frac{xy}{x + y} = \frac{x(x^4 - x)}{x^4} = x - \frac1{x^2}$$