One way to explain that division of $x$ by $0$ is undefined is by contradiction. Suppose $x/0 = a$ and suppose $x$ is a non zero value. Then, by cross multiplication, we get $0\cdot a = x$. At this point ask the child what number times $0$ equals a non zero number. After a little thought the child will most likely say that any number times zero is $0$ so that $0\cdot a = x$, $x$ a non zero number is not possible. Next consider $x = 0$ so you have $0/0$. Let $0/0 = b$, where $b$ is a non zero number. Then you cross multiply to get $0\cdot b = 0$. Now ask the child to come up with a number that satisfies this equation. The child will most likely realize that any number will do and pick one, say $5$. $0\cdot 5 = 0$, true. Now say, what about $0\cdot 6$? The child will say that equals zero too. So, going back to $x/0$, there is no solution and in the case of $0/0$, in effect, any solution will do. Neither of these are allowed in mathematics. The above is not a proof of course but it might help a little. Note: the explanation doesn't really work for the case where $x/0 = 0$ or $0/0 = 0$. I imagine this observation would have to be modified a lot to be useful but perhaps it would be a good starting point for explaining that division by $0$ is undefined.
Also, a way I use to think of limit is to imagine what happens when you place smaller and smaller number under, say $1$. $1/(1/2)$, $1/(1/100)$, $1/(1/1000000)$ etc. I imagine that any child will know that you flip the denominator to simplify these equations so you get $1\cdot (2/1) = 2$; $1\cdot (100/1) = 100$; $1\cdot (1000000/1) = 1000000$ etc.
These two approaches combined might be used to explain one reason for limit. One thing that limit allows is to go as close as you would like to something that is not possible, ie. division by zero.