Division by zero can be a difficult concept to grasp, and with good reason. Division is basically dividing the dividend into as many groups as the divisor. But with zero, you can't do that, because you cannot divide a whole number into zero groups.
Another way of looking at this problem is, if number "A" is divided by number "B", then the result is number "C". Also, number "C" multiplied by number "B" MUST equal number "A". However, this presents a logical impossibility. If number "B" is zero (which it is in division by zero) and number "A" is a whole number, then "C" multiplied by "B" CAN'T equal "A", because any number multiplied by zero is zero (if A≠0 and A/B=C then C*B=A and if B=0 then C*B≠A because C*0=0). This presents the problem with dividing by zero (the result of division by zero is impossible to give a value to).
If the equation 1/x is graphed, the graph has an infinite discontinuity as you approach zero. If you take the limit of the equation as you approach from the left (x is getting larger), y approaches negative infinity. However, if you take the limit as you approach from the left (x is getting smaller), y approaches positive infinity. This concept - that at zero, 1/x is basically undefined - means that division by zero doesn't exist!
Another possible way to look at this problem is using the concept of a remainder, which is taught in elementary schools across the nation. If a number cannot be equally divided by another number, simply find how many times the divisor goes into the dividend without going over the dividend, then find the difference between the dividend and the closest multiple of the divisor smaller than the dividend (for example 13/3 3*4=12 13-12=1 so 13/3=4 R 1). This concept isn't commonly used in higher level math, however it can be valuable in the concept of division by zero. For example, if 3 is divided by 0, the result would be "x" remainder 3 (3/0=x R 3). I used "x" because that number could be any number, and the equation would still work (see the previous explanation of the concept of remainder). This means that the value of any number divided by zero is not a fixed number, but rather an impossible concept.
In conclusion, division by zero fails to pass the basic requirements of division in general, and as such is not possible.