Ending points of the root locus Let $$D(s) + KN(s) = 0 \tag{1}$$where $D(s)$ and $N(s)$ are polynomials of $s \in \mathbb{C}$ such that $\text{Deg}(D) = n, \ \text{Deg}(N) = m$ and $n\ge m$. The root locus method tells us how the solutions of $(1)$ changes as we change parameter $K$ from $K=0$ to $K = \infty$.
I'm trying to understand these extreme cases. Let $K \to 0$ and we have $$D(s) = 0$$ so in this case, the set of the solutions is $A = \{s \in \mathbb{R} | D(s) = 0\}$.
Now let $K \to \infty$, if we choose $s$ such that $N(s) \not = 0$ then the answer will be infinity. So we should choose $s$ such that $N(s) = 0$. In that case, if $(1)$ holds, we also have $D(s) =0$ which means $N(s)$ and $D(s)$ have the same factor but this isn't the result that should be obtained. Curiously, if we rewrite $(1)$ $$\frac{N(s)}{D(s)} = -\frac{1}{K} \tag{2}$$ and let $K\to \infty $, one possible case that $(2)$ holds is $N(s) = 0$ which gives us $m$ solutions and this doesn't require $D(s)$ have the same factor as $N(s)$! Why this happens? And why the first solution is wrong?
Example: Let $D(s) = s^2 - 4$ and $N(s) = s + 3$. So $(1)$ becomes $$s^2 - 4 + K(s+3) = 0$$If $K \to \infty $ and $s = -3$ then $9 - 4 = 0$ which is clearly wrong. On the other hand, rewriting the equation $$\frac{s+3}{s^2 - 4} = -\frac{1}{K}$$ If $K \to \infty $ and $s = -3$ then $\frac{0}{9-4} = 0$ which is true, of course.
The problem with your example is that $\infty\cdot 0$ isn't necessarily equal to zero. The only way to judge what is happening in the limit $K\to\infty$  is to divide the original equation by $K$:
$$\frac{D(s)}{K}+N(s)=0\tag{1}$$
Now it is obvious that for $K\to\infty$ the actual value of $D(s)$ is irrelevant, as long as it is finite. Consequently, the only necessary condition for $(1)$ to be true as $K$ becomes large is that $N(s)=0$.
This problem is a bit similar to the problem of determining the limit
$$\lim_{x\to\infty}\frac{x}{x+c}\tag{2}$$
The value of the limit $(2)$ is independent of the choice of the constant $c$, which becomes negligible compared to $x$. The same is true in the first equation of the question: the actual value of $D(s)$ becomes irrelevant compared to $KN(s)$.