# Infinity As a Limit

The most likely source for the question whether 1/0 = ∞ is a realization that dividing 1 by ever smaller (real) numbers produces numbers arbitrary large. In this context, ∞ is understood as a very big, in fact, even bigger than any other, number. In a sense, this is a good idea that may be worked out rigorously. However, the approach is not without pitfalls. One can say that ∞ is more "big" than "a number". This is because, no definition may make ∞ possess properties of (or behave like) all other numbers.

For example, assuming that indeed 1/0 = ∞, we should also accept (by exactly same reasoning) that

On the up side, if 1/0 = ∞, then it is quite likely that 1/∞ = 0. Indeed, if we understand that 1/∞ is a substitute for dividing 1 by ever larger numbers, then 1/∞ may sensibly stand for a non-negative number which is smaller than any positive number; and 0 quite fits the bill.

There is one problem, though. 1/0 is an ambiguous expression. Somehow,

So what is ∞?

First of all, it is just a symbol for the concept of *growing without bound*. Instead of saying "let x (or n) grow without bound", mathematicians often say "let x (or n) tend to infinity" or "as x (or n) tends to infinity". There is a special shorthand for this, too:

As x → ∞, other quantities that depend on x, like say, f(x), may exhibit all kinds of behaviors. Some, like

f(x) → ∞ as x → ∞,

or, introducing another symbol *lim*,

lim f(x) = ∞ as x → ∞.

And also

lim_{x → ∞} f(x) = ∞.

A quantity, f(x), dependent on x may grow without bound as x tends to a real number as well. In this case, we write

lim_{x → a} f(x) = ∞,

where a is a plain real number. In particular, if f(x) = 1/x, we would like to write

lim_{x → 0} 1/x = ∞.

However, as we already discussed, this rather meaningless, for 0 may be approached from two directions producing quite distinct results. Instead we use

lim_{x → 0+} 1/x = ∞ and

lim_{x → 0-} 1/x = -∞

to distinguish between the two cases.

It is important to realize that none of the above makes ∞ a (real) number. In the real number system, 1/0 is quite meaningless, or, at best, ambiguous. Limits are studied at the beginning Calculus courses where it is shown that if

lim_{x → a}f(x)g(x) = AB = (lim_{x → a}f(x)) (lim_{x → a}g(x)).

However, taking g(x) = x, h(x) = x² and k(x) = √x, all of which grow without bound as

lim_{x → ∞} f(x)g(x) = 1,

lim_{x → ∞} f(x)h(x) = ∞, and

lim_{x → ∞} f(x)k(x) = 0.

This tells us that the expression 0·∞ will forever remain undefined.

The addition of limits is handled similarly:

lim_{x → a}(f(x) ± g(x)) = A ± B = lim_{x → a}f(x) ± lim_{x → a}g(x).

As with the product, it is not always possible to use that formula with infinite limits. An expression,

lim_{x → ∞}f(x) = lim_{x → ∞}g(x) = ∞.

However, the difference f(x) - g(x) = sin(x) has no limit as

For those curious, the symbol ∞ for infinity was borrowed from the Latin numeral 1000 by John Wallis in 1655. The symbol ∞ closely resembles the shape of *lemniscate* - a simple *endless* curve.

- What Is Infinity?
- What Is Finite?
- Infinity As a Limit
- Cardinal Numbers
- Ordinal Numbers
- Surreal Numbers
- Infinitesimals. Non-standard Analysis
- Various Geometric Infinities
- Paradoxes of Infinity

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