This node assumes that you already know what a limit is. It also assumes your browser is capable of displaying various mathematical symbols, which most are. If yours is not, I apologize in advance.
The funny thing about infinity is that it doesn't like to go away, or change, except under extraordinary conditions. Outside of these exceptions, any mathematical statement that has ∞ in it is likely to equal ∞. That's just the way things go. Let's have a run down of the basics, first.
On things equaling infinity: RPGeek correctly reminded me that saying things such as "the set of all integers is equal to infinity" is misleading. More accurately, it would be said that the size of the set of all real numbers is infinite. While this is in fact more accurate, in the following paragraphs, I will stick with the less accurate but easier to say method. This write up would become far to cumbersome otherwise, methinks.
To be perfectly clear right here and now, however: Infinity is a word that indicates endlessness.
Where c is any constant, real number
∞ ± c = ∞
Yes, this means that ∞ + 1 is no bigger than ∞, sorry to disillusion your childhood there. For that matter, ∞ - 1 is no smaller than ∞ either.
∞ ± ∞ = ∞
This may come to a surprise to some of you, who are still thinking that ∞ is a discrete number that is in any way entitled to obey the rules that every real number in existence obeys. Get over this idea fast. Consider the following example: take the set of all real integers, which equals ∞. Now remove all the odd numbers. Since the set of all odd numbers also equals ∞, you have just subtracted ∞ from ∞, and yet you still have the set of all even numbers left, which is also equal to ∞. Going the other way, if you take the set of real integers and add all the halves in, you get a set which is still just ∞, no bigger or smaller than ∞
In a way, when we're subtracting all the odd numbers from the set of all real integers, we're really just crossing out ever other number, which is the same thing as diving by two. And the reverse, which is the same thing as doubling. Which leads directly into:
∞ / c = ∞
∞ * c = ∞
But it gets better. Going back to the example of adding in all the halves in between, so the set would go 1, 1.5, 2, 2.5, ..., ∞, you could continue on by adding all the thirds, and fourths, and fifths, and sixths, and so on forever, and yet still equal infinity. As such:
∞ * ∞ = ∞
∞ / ∞ = ∞
Please note that the above does not include any limits. Limits are covered later in the node. For now, we're talking conceptually about pure infinity in the manner that I described above.Anyway, it follows from this that...
∞c = ∞
Now that we've gotten that out of the way, onwards to the
exception. What statements involving ∞ do not simply equal ∞? Well, try this example:
∞∞ > ∞
I will not be dealing with this in this node. Go look up
transfinite numbers,
aleph-null, or even
omega-null if you are interested. What is more useful for the topic of the node is the following:
c / ∞ = 0
Well, to be honest, that
statement isn't 100% factual, although it is a way of informally stating something completely different that happens to be true. The concept of
dividing a finite thing into
infinite portions isn't quite sensical¯ but we can approach the concept using limits, which will do the job nicely. The more accurate statement is thus:
lim c/x = 0
x→∞
There is a
proof for this, which goes ∀ε>0∃N>0∋x>N⇒|1/x|<ε
I'll leave the sorting out of that as an exercise to the reader, or for
another node.
Lets look at a little example, and go through it
intuitively and then a little more
rigorously.
lim (x2-1)/(x2+1) = ?
x→∞
Solving this the
intuitive way, we can just
substitute a really really big number for x, and see where the
limit approaches.
ƒ(1020) = (1060-1)/(1060+1)
And since our x here is so large, it dwarfs the constants involved so much that we can approximate said constants as not existing.
ƒ(1020) = 1060/1060 = 1
A more rigorous proof goes as follows:
lim (x2-1)/(x2+1)
x→∞
lim (x2-1)/x2
= x→∞ ---------
(x2+1)/x2
lim 1 - 1/x2
= x→∞ --------
1 + 1/x2
And since 1/xc = 0,
lim 1 - 0
= x→∞ ----- = 1
1 + 0
Many problems involving an infinite limit and a ratio can be solved in a similar manner, but sometimes you'll come across nastier little buggers, with various degree polynomials on the top and bottom. Fortunately, there is a very easy shortcut, which even applies to the above problem.
Suppose P(x) is an nth degree polynomial, and Q(x) is an mth degree polynomial. Then let ƒ(x) = P(x)/Q(x)
Pnxn + Pn-1xn-1 + ... + P2x2 + P1x + P0
ƒ(x) = ------------------------------------
Qmxm + Qm-1xm-1 + ... + Q2x2 + Q1x + Q0
If n=m, then...
lim P(x)
x→∞ ----
Q(x)
lim Pn + (Pn-1)/x + (Pn-2)/x2 + ... + P0/xn
= x→∞ -------------------------------------
Qn + (Qn-1)/x + (Qn-2)/x2 + ... + Q0/xn
And because c/∞c = 0...
Pn
= --
Qn
What this means is that if you're trying to find the infinite limit of a function that consists of a
ratio between two polynomials of the same degree, all you have to do is look at the leading two numbers. For example:
lim 3x5 + 84x4 + 32x3 + 2x2 + 67x + 65243
x→∞ ------------------------------------
2x5 + 56x4 + 5x2 + 3432
Is just equal to three halves. I can tell this with only the barest of glances. How? First I note that the highest power in both the top and bottom is 5. The top
coefficient to this highest power is 3, the lower is 2. Everything else is ignored. If you feel like taking the time, you are welcome to solve this problem the long way, like I did on the earlier problem, but I'd really rather not, thanks.
On the other hand, if the degree of the polynomials is NOT the same, the answer is even easier.
if n>m...
lim P(x)
x→∞ ---- = 0
Q(x)
and if n<m...
lim P(x)
x→∞ ---- = ∞
Q(x)
There's one last special case, and that's if n=m+1. This obeys the previous rule, so the limit will be equal to ∞, however, the function will
converge to a
straight line as x gets larger and larger. This is called a slant asymptote. You can find a graph of the line it converges to like this:
if n=m+1
lim ƒ(x) = kx + b
x→∞
Some algebra will enable you to find k and b.
Now, while limits to infinity are useful for finding asymptotes of functions when curve sketching, they become significantly more useful when attempting to determine an infinite sum, and in particular, a riemann sum.
References:
Asimov, Isaac "Asimov on Numbers", Copyright © 1977 by Isaac Asimov
Stewart, James "Single Variable Calculus, Fifth Edition", Copyright © 2003 by Thomas Learning Academic Resource Center