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 → ∞.
limx → ∞ 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
limx → a f(x) = ∞,
where a is a plain real number. In particular, if f(x) = 1/x, we would like to write
limx → 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
limx → 0+ 1/x = ∞ and
limx → 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
limx → af(x)g(x) = AB = (limx → af(x)) (limx → ag(x)).
However, taking g(x) = x, h(x) = x² and k(x) = √x, all of which grow without bound as
limx → ∞ f(x)g(x) = 1,
limx → ∞ f(x)h(x) = ∞, and
limx → ∞ f(x)k(x) = 0.
This tells us that the expression 0·∞ will forever remain undefined.
The addition of limits is handled similarly:
limx → a(f(x) ± g(x)) = A ± B = limx → af(x) ± limx → ag(x).
As with the product, it is not always possible to use that formula with infinite limits. An expression,
limx → ∞f(x) = limx → ∞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