This is part of a series on common misconceptions.
Is this true or false?
Why some people say it's true: We have and with all positive real values of . Therefore
Why some people say it's false: Based on the 's property, we know that . Therefore cannot be larger than
The statement is .
Proof: You may say that , but that means we have and so
We cannot do ANY arithmetic operator to infinity. It is not legal to do so since is a concept, not a number. It is also uncountable. Therefore we cannot compare and
Rebuttal: Then what is in the school?
Reply: That is only when you evaluate sum of the limits of and altogether. If both of them are infinity, then the sum will end up as .
Rebuttal: But if you cannot do any arithmetic operator with limit, how comes ?
Reply: Let's put it in this way: > 0 and < 0. A number greater than 0 multiplied by a number less than 0 will result in a negative number; therefore, a really large positive number multiplied by a really large negative number will end up with a really large negative number.