Now even Newton was fuzzy about the limit (not about much else!), and this led to a very rare event in Western History. The event is that a contemporary of Newton, an exceedingly bright philosopher/theologian named Bishop Berkeley, correctly rapped Newton's knuckles about what Newton called "infinitesimals" (sorry, but that's a very long story), so let's try to be "intuitive" about the limit, even though that's often a mathematically slippery way to avoid really taking care of business.
Here's an example. Imagine the following function: y = f(x) = 1/x, and then ask yourself what does 1/x "get infinitely close to" as x "approaches infinity”.
There are all kinds of fancy notation for this, but I'm deliberately avoiding it since I don't want to reinforce what I believe to be the most common misunderstanding of mathematics. And for a change, this is easy to put into words.
*drum roll*
The most common misunderstanding of mathematics is confusing the "reality" of mathematics (a big and much argued topic) with the "symbology" of mathematics. And I confess when I was learning this subject, I too fell prey "quasi consciously" to that belief. After all, the notation was so impressive and "different" that it seemed like what I saw on the page WAS mathematics.
However, in due course I outgrew that belief (as do nearly all mathematicians -- I assume), and realized a better way to understand the symbology was to see it much the same way you look at musical notion. After all, no one thinks that notation "is" music, because they understand musical notion is basically "directions" for you to follow, and if you follow the directions well, "something happens" which we call music.
Actually, almost EXACTLY the same thing is true for mathematical notation. The notation is telling you something to do "in your head" and if you carry out those instructions properly, you will hear (can we be poetic again!) "mathematical music". In short, the notation is not the "real thing"; rather they are instructions to follow so you have the opportunity to experience the real thing. It's very intellectually liberating to get this straight, so you stop (in the words of my former high school debate coach) putting the emPHAsis on the wrong syLLAble.
Well, I keep going off on tangents, don't I? I guess my only excuse is that this is fun for me and I increasing suspect I really don't WANT to get back to politics and all that b.s.
However, the limit thing is so beautiful and powerful; I'm determined to make a connection between it and the world we wake up to in the morning.
Here's an "intuitive" definition of the limit. It's giving you an exact description of a place you'll never reach, and even though that should be qualified, let's run with it anyway. For example the limit of 1/x as x goes to infinity, is zero (we never said what it was before). Think of it like a plane in a science fiction movie that "approaches" the runway, and indeed gets unimaginably (literally) "close" to the runway, but never makes contact.
You see, that is what is happening to 1/x as x keeps getting larger and larger. 1/x will NEVER "equal" 0 (in fact, 1/x = 0 is a contradiction because it has no solution), but the LIMIT of 1/x as x gets huge is zero. When x equals a billion, you can take it to the bank that 1/x is very, VERY close to zero -- even though it is not and never can be zero.
Finally, the limit concept is the soul of calculus and since calculus was the supreme mathematical tool of science for a couple of centuries, it is clearly a very "deep" concept. Sometimes I tell my students the limit is not the Cheshire Cat; it's more like the "smile" of the Cheshire Cat. For those of you who know calculus, you also know that differentiation and integration are simply two different applications of the limit.
Hence, all roads in calculus lead to the limit -- which, notice involves infinity ("infinitely close") which is the real reason it is such a profound intuition.
OK, time to relate the "limit" to human, social existence.
Arguably, politics is not only more part of the problem than the solution, it IS the problem, since it promises much, but always delivers next to nothing. Said more exactly, "politics" is the social/political game created by the elites to seduce and control the human race, and thus it all comes down to two choices:
(a.) We can "play" this game (and ALWAYS lose, since all the game rules were created by the elites). For example, for many (most?) of the people who elected Barrack Obama, it looks very much like the cards were stacked (again!) and we're getting, not the campaign rhetoric Obama, but an elite bought and paid for Obama. However, he gets his hundred days, and then we will know everything.
