There is a mathematical concept known as the asymptote. This concept is taught to most students in the second or third year of high school math. An asymptote refers to a straight line that a graph approaches but never touches; later in calculus or at the end of pre-calculus the similar concept of a limit is introduced.

Simply put, the fraction x/0, x being any real number, is an example of an asymptote or limit. However, it is unlike defined limits, such as the limit of the function f(x) = a^-x as x approaches infinity. In the case of f(x) = 2^-x, for instance, the value of the function approaches the definite value of zero as x gets larger. This is not the case with something like x/0, because infinity is not defined. Calculus tells us the limit of x/0 is infinity, but infinity is not a number. It is a mathematical construct used to represent the concept of a number larger than all other numbers.

If you read that last sentence and think about it, you’ll see the problem. Let’s suppose, for instance, that a famous mathematician tells you, “The largest number in the world is 30,000,000!” You can reasonably ask, what about 30,000,001? The mathematician may then respond, “Yes, 30,000,001 is the largest number in the world, as well as 30,000,000!” Later on in the evening, the famous mathematician will laugh about this with his friends over beer and poker. But that’s not the point. The point is, for any number one can declare to be the largest, there will always be a way to obtain a larger number than that; you can simply add one. This is the essence of infinity, and the reason infinity is undefined.

If you divide a number by 0.000001, you will get quite a large number as a result. The closer the denominator gets to zero, the larger the result of the division. This means that the limit diverges; it does not settle near a single point. The closer you get to zero on the bottom, the more you spiral out into infinity.

In short, division by zero yields a result of positive or negative infinity. It is thus impossible to divide by zero and get a defined result. If you can accept infinity as a valid result, then you can reason that dividing by zero is possible. It is unequivocally clear, however, that division by zero is undefined.