It is undefined because the inverse of division is multiplication. If you multiply by zero, every answer is zero. If you try to invert that operation you can't know which number was multiplied by zero to get zero because multiplying by zero doesn't produce a unique answer for each operation.

Additionally, if you take the limit of 1/x as x approaches zero from the positive side the result approaches positive infinity. If you take the limit from the negative side it approaches negative infinity.

An interesting thing to think about is whether multiplication by zero really makes much sense in the concrete world. You can't really have zero groups of something or some number of groups of zero. Zero groups of anything is still nothing. We can think of that abstractly once we have the abstract concept of numbers, but in the real world that idea is nonsense.