For the record, I can't help but read the title in my head with a badly rendered Scottish accent. Moving on, I've decided to start some actual calculus in my quest for calculus. Our first topic will be limits which are fundamental to the understanding of calculus and higher math in general. Absolutely necessary might be a better description. The fact of the matter is, without limits, you have no calculus.
I've struggled with how to introduce limits. I've run into a chicken or the egg problem. The chicken being what limits are used for and the egg being how to interpret them. Long story short, evolution suggests that the egg came first and though I chose the order in my analogy rather arbitrarily, I'll hold to it and begin with the interpretation of limits. For the time being you'll have to take my word for it that they're important.
So, we'll start with the syntax:
The previous statement reads: The limit of
as
approaches
is equal to
. Alternatively, this can be interpreted as: The values of
get closer to
as
gets closer to
. Further, as
gets arbitrarily close to
, the value of
gets arbitrarily close to
(arbitrarily close in math terms suggests there is no difference between the two values). Our initial investigation into limits will be by example. We'll then move into more mathematically verbose methods.
The most familiar piece of the lexicon is
, so it seems like a good starting point. We'll start with a function, pick a value for
, then work toward finding a value for
. Here we go:
Let
. This gives us
which we know is equal to
. Now we just have to figure out what
is.
Calculating
for some values of
and we get:
Going back to our limit,
, and looking at the the data, we can start figuring out what is meant by: as
approaches 0. Starting with -4 and moving towards 0 (from left to right on the number line), we can see that
is getting smaller. We can also approach 0 from the right. In this case we start with 4 and move from right to left towards 0. Again, we notice the value of
getting smaller. In fact, the function
is defined at 0 and as such we can see intuitively that as
approaches 0,
gets closer to and eventually equals 0. With that,we can say with some semblance of confidence that
. This is by no means proof of the limit(we'll get to that I promise), but it does give us a starting point.
With our very basic example, it is very tempting to think that we can find
by simply substituting the value of
into
suggesting
. Although this works here, it is a fallacy to assume that this technique works for all limits. Take for example
Plugging in the value of
which in this case is 2, we get
which of course is
. I assure you, that's not the right answer. So, assuming that plugging in the value for c doesn't get us where we're going, we have a few standard approaches to try. In no particular order we have factoring, multiplying by the conjugate (in the case of a radical), expand (just like factoring but backwards), and of course magic. There are other techniques, one of the most helpful being
L'Hopital's rule, but as that requires derivatives to use and we need limits to understand derivatives, we'll skip it for now.
Back to the problem at hand. What do we do with
? I don't see any radicals, we don't know enough to use L'Hopital's rule, and there is nothing to expand. That leaves us with magic and factoring. Since we don't want to abuse the magic card so early in the game, we'll stick with factoring. With some simple algebra, we can determine that
. Plugging this back into the problem we get
this quickly simplifies to
. Now we're in a much better position to plug in the value for
giving us
.
From the top, we get:
Suggesting that
Again, this is not a proof. That's not to say that it's a coincidence either. That is in fact the limit of the function as
approaches 2. We just need something a bit more solid mathematically to actually prove that it's correct. This post is just an introduction to limits, we'll cover the proofs in the next one. Settle down, you can wait, I promise.
