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 2/0 = ∞ implying that 2×∞ = ∞. Obviously, this is a property that is not shared by any real number. Similarly, ∞ + 1 = ∞ which after 2×∞ = ∞ should not come as a surprise for the latter may be expected to mean ∞ + ∞ = ∞. So if adding another infinity does not change it, adding a mere 1 should not change it either.

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, 1/∞ = 0 makes more sense than 1/0 = ∞. The reason is that, the way it was used so far, ∞ may be approached from only one direction, viz. by letting a number grow without bound. Zero, on the other hand, may be approached from two directions. If 1 is divided by ever decreasing numbers the result grows without bound, as expected. But zero is exactly midway between positive and negative numbers and may be as easily approached by negative numbers decreasing in magnitude. The result will be a negative number whose magnitude grows without bound. This one is more appropriately denoted as -∞.

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: x → ∞ (or n → ∞).

As x → ∞, other quantities that depend on x, like say, f(x), may exhibit all kinds of behaviors. Some, like f(x) = x², will grow without bound. In such cases, we write

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 f(x) → A and g(x) → B, as x → a, then

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 x → ∞, and f(x) = 1/x, we see that

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, ∞ - ∞ (in the sense of the difference of limits) may happen to evaluate to -∞, a finite number, or ∞ depending on the two limits involved. For f(x) = x and g(x) = x - sin(x), both limits are infinite:

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

However, the difference f(x) - g(x) = sin(x) has no limit as x → ∞. Thus the expression ∞ - ∞ will also remain undefined.

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.

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