I'm a fifteen year old currently studying calculus and our teacher today explained the properties of infinite limits. Firstly, these were intriguing, since they appeared to imply that one can do algebraic manipulation with infinity, something I've long been told cannot be done, because "infinity is a concept, not a number." One of the properties was particularly interesting:
If lim(f(x)) = infinity, and lim(g(x)) = L, then the following is true: x->c x->c
Lim(g(x)/f(x)) = 0 x->c
Now, this basically is saying that the limit of 1/infinity = 0, which makes sense and works with both positive and negative infinity (1/-1000000000 is very close to 0, therefore the limit of 1/-infinity should also be equal to 0). Therefore, I introduce a new function h(x), whose limit is thus:
Lim(h(x)) = -infinity x->c
Using the property mentioned,
Lim(g(x)/h(x)) = 0 x->c
Now, since both limits are equal to 0, they are equal to one another. (a=b, b=c, thus a=c)
Lim(g(x)/h(x)) = Lim(g(x)/f(x)) x->c x->c
Using the quotient property of limits (Lim(x/y) = lim(x)/lim(y)), we get this:
Lim(g(x))/Lim(h(x)) = Lim(g(x))/Lim(f(x)) x->c x->c x->c x->c
Cross-multiplying yields the following result:
Lim(g(x)) * Lim(f(x)) = Lim(g(x)) * Lim(h(x)) x->c x->c x->c x->c
Using the original definitions of these limits, in which the limit of f(x) at x-> c is infinity, h(x) at x->c is -infinity, and the g(x) at x->c is L, we get:
L * infinity = L * -infinity
And since anything times infinity is just infinity, we get:
infinity = -infinity
This startling result was very interesting. I don't think I made any mathematical mistake, so the only two things that I can think of that fix this issue are the following:
- The limits of 1/infinity and 1/-infinity are not both 0, rather the 1/infinity is +0 and 1/-infinity is -0.
- The limits are correct and infinity = -infinity, in which case the number line is a circle and infinity and -infinity are the same.
Help and critique on my logic is requested, as well as an explanation from a more experienced mathematician.