Zero

Numerous teachers of mathematics will tell you that x/x = 1, (x+1)/(x+1) = 1, (x-7)/(x-7) = 1, etc.

This is not quite correct. If x = 0, then x/x does not equal one. There is no inverse for zero and hence 0/0 has no answer.

In the second example above if x = -1 we again have 0/0. In the third example if
x = +7 we again have 0/0.

Hence, (x +9)/(x+9) = 1, but x cannot equal -9 is the complete answer.

Suppose you graph y = (sine x)/x. When x = 0 we have y= (sine x)/x = 0/0. Hence at x = 0, y = 0 on the number plane we put in a discontinuous point.