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.
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