Another way to illustrate why division by 0 is undefined is to study the graph of By using a table of values some points on the curve can be found.
Graphing the points on a coordinate plane allows the behavior of the function to be shown. The red points show what happens as x approaches 0 from the left, and the green points show what happens as x approaches 0 from the right.
The closer x comes to 0, on both sides, the more extreme the values of the functions become. The quotient approaches when x approaches 0 from the right, and when x approaches from the left. Because is a function, f can only produce one value when x=0. Therefore, division by 0 is undefined.
Since both sides of the equation are divided by a−b=0, and a−b=0, the expression was validly manipulated to create a false statement.