Thread: Steven's Trilogy - Zero Equals Infinity

Well of course it's not a number. However the limit still exists and it is still infinity.

The function I gave has a limit as x --> 3^-. It is an undefined number, or infinity.

'Infinity' is an expression of the lack of a limit though.

I think the key point is that the limit of a function towards a number doesn't imply things about the function at the number. The question is 'what is 1/0'; the fact that the limit of 1/x as x tends to 0 is infinity does not imply that 1/0 is infinity.

The problem with such a statement is clear if you consider what happens when you take the limit in different directions; you get plus and minus infinity. They can't both be the value of 1/0.

Why not? Both require infinitely large numbers to "work". Just as dividing any constant by infinity is basically zero as the result is "infinitely" small, the only number which would work is 0.

At your last point; that depends on the function. f(x) = (3)/(3-x)² has a 'limit' at 3.

But dividing any c by minus infinity would also be 0, so c/0 = -infinity

1/0 can't be two different things. Then division would no longer be an operation.

f(x) = (3)/(3-x)2 still doesn't have a value at 3 because as I've explained limits don't imply things about functions at the limit.

Originally Posted by Xei

But dividing any c by minus infinity would also be 0, so c/0 = -infinity

1/0 can't be two different things. Then division would no longer be an operation.

Why can't it be two different things? That only works with numbers, and as infinity is not a number, it should naturally behave differently.

Because in mathematics we want there to be a single answer to 'what is 1/0'; otherwise we get useless systems or contradictions.