Think about the graph of 1/x. As x tends to 0 from the right, sure, it tends to +infinity. But if x tends to zero from the left, it tends to -infinity. So, by the logic you've used, at x = 0, 1/0 = +- infinity - there is no definite value.
Alternatively, you can say that if x/0 = y, x = y*0, so 0*y =/= 0, which is incorrect. Hence we simply cannot divide by zero.