This is part of a series on common misconceptions.
Is this true or false?
Why some people say it's true: Any number divided by itself is 1.
Why some people say it's false: We cannot just do arithmetic with something that is not a number.
The statement is .
Proof: We know we cannot do arithmetic with infinity. But let's take a limit and see if it is true:
We know two such functions are and . But the limit is then 2 and not 1, and hence it is not necessarily 1. The limit is multivalued and is undefined.
Rebuttal: If , then .
Reply: You are cross multiplying, but it is not legitimate here. Let's multiply both sides with . We get . Then you assumed that the infinities would cancel out to one, but remember they are not 1.