negative and positive infinity

Question

This is a weird question that I thought of and I was wondering if I could get some help.

So normally $\frac{1}{x} = \frac{1}{y}$ then x and y would have to both be the same number, but with infinity $\frac{1}{-\infty} = \frac{1}{\infty}$ because 0 = 0 . How does this work? is the answer not really 0 for both but instead approaches 0?

Answer 1

Since $\infty$ isn't really a number it doesn't exactly make sense to say $\frac{1}{\infty}$.

But one could probably say, intuitively

$$\frac{1}{\infty} := \lim_{n\rightarrow\infty} \frac{1}{n}$$

and similarly,

$$ \frac{1}{-\infty} := \lim_{n\rightarrow\infty} \frac{1}{-n}$$

In this case both are zero as you claimed, but this doesn't imply $\infty = -\infty$ because these aren't real numbers you can manipulate in the usual way, just formal symbols.

Answer 2

For all numbers $x$ and $y$ with a multiplicative inverse, we can say that $\frac1x=\frac1y$ if and only if $x=y.$ However, $\pm\infty$ are not numbers. Rather, they are used to indicate increasing/decreasing without bound, a notational convention only. Attempting arithmetic with $\pm\infty$ isn't a good idea, because there are so many things that can go wrong. Some of the answers here discuss potential pitfalls of doing this (and the question, itself, is a very illustrative example).

On the closely related topic of division by $0$, you can find more here, in the linked questions, and in the questions that those questions are linked to.

Amusingly, arithmetic with the so-called "point at infinity" in the extended complex plane is less problematic. See here under Real Projective Line and Riemann Sphere.

Answer 3

For some purposes it makes more sense to speak of a single $\infty$ that is approached by going in either the positive direction or the negative direction than of two separate entities called $\pm\infty$.

And for some purposes it doesn't. The function $\dfrac{1}{1+2^x}$ certainly approaches $0$ as $x\to+\infty$ and approaches $1$ as $x\to-\infty$.

But when you're talking about rational functions $f$, you have $f(x)\to\text{something}$ as $x\to\infty$, then the "something" is the same regardless of whether $x\to+\infty$ or $x\to-\infty$. (In particular, if one has a slanted asymptote and $f(x)\to+\infty$ as $x\to+\infty$ then $f(x)\to-\infty$ as $x\to-\infty$, but one can say $f(x)\to\infty$ as $x\to\infty$ and construe both instances of $\infty$ as the single $\infty$ at both ends of the line.) And one has $\dfrac{5}{x-8}\to\infty$ as $x\to 8$, and one need not distinguish between approaching $8$ from the right and approaching $8$ from the left. That makes rational functions everywhere continuous.

So also with values (i.e. outputs) of trigonometric functions, but arguments (i.e. inputs), go from $0$ to $2\pi$ and regards $0$ as the same point as $2\pi$. Then $\tan x\to\infty$ as $x\to\pi/2$, and that's just $\infty$ rather than $\pm\infty$ and one need not worry about which direction $x$ approaches $\pi/2$ from. This makes $\tan$ and the other trigonometric functions everywhere continuous.

This way of viewing things also fits well into projective geometry.

So $\dfrac1x\to0$ as $x\to\infty$ and $\dfrac1x\to\infty$ as $x\to0$, and the reciprocal function is everywhere continuous and is one-to-one.