Okay, now we're gonna start talking about
limits here a little bit. So limits are one of those things they're just
not completely
mechanical like we would like them to be. So you have to know some different
tricks on how to deal with them.
So to begin with, let's start off with this equation "y = x - 4".
The basic idea with the limit (we can
use an example to explain as follows), so
suppose we take numbers
close to 5. "5.1
5.01, 5.001"
or say "4.9, 4.99
4.999". Well,
if you did the arithmetic with those numbers,
in each case you would get out a number
relatively close to the number one.
And that's basically the fundamental idea of a limit.
We're putting in numbers close to a specific number
and we're just trying to see what numbers we would get out of our
formula.
The notation for this is (we write
in this case) we write "the limit as X approaches 5"
I'm taking x coordinates close to five and if I plug them
into the formula, I get numbers close to one.
Now I think this notation's a little deceiving because
here there's a little arrow that says as "we take numbers close to five
it says the limit equals 1" and it's a little deceiving.
What you should really think is "as X gets close to five
X - 4 is getting close to the number one".
But the notation is what it is and they use an equal sign.
But I think this is something that confuses people and
they see this equality and they think "hey, it's definitely equal".
Okay, in general we write "the limit as X approaches a"
so we're taking x coordinates close to something we're plugging them into our
function and we get some number out.
the number out that we get is known as
"the limit of the function". Okay so
there's definitely a whole lot of different ... different ways that you deal
with limits.
So, to start off with, there's a couple different cases
so we have the limits where X approaches a finite number (like in our last case)
or X could be going off to positive or negative infinity.
And all that means is we're putting in either very, very large positive numbers
into our formula we're trying to figure out what we get out
or putting large negative numbers
into our formula and we're trying to see what types of numbers we get out.
We'll deal with this case
at the beginning. Okay, so there's a couple things that can happen
the basic idea on how you try to do a limit
is you "plug and chug". So, for example,
in our last example, notice if you simply plug in
"X equals five" well 5 - 4
is one and that's your solution. Typically,
most limit problems don't work out as easy as this though.
Typically when you plug in "X equals the number"
into your function, you'll usually either get something of the form
0 on top and 0 on the bottom (this is definitely not 1)
or you get something non-zero over 0.
We're gonna work on some of the cases where you have 0 over 0
and again, these aren't hard-and-fast rules but they are general rules.
So if you have something of the form "0 over 0",
typically your solution will end up being a finite number,
at least for a lot of the examples that you'll encounter. And there's probably
four good common tricks to always use to approach these problems.
One is you can try to factor and cancel. If there's a fraction floating around
try to get common denominators and simplify it
down. If there's something to multiply out, try to expand it and multiply it out
and see if that helps.
If there's a radical involved, try multiplying by the conjugate
and then there's the "others..." just kind of
problems that don't fit into one these molds and you may have to be a little
creative on them.
So take a look at one of the other videos. I'm definitely going to do an
example of
all of these and then we'll come back to the limit at infinity
and also what we do on the non-zero over zero cases.
Again, the limits are frustrating so if they're tripping you up, you're definitely in
good company.
I think everybody tends to have a little bit of an issue with them at the
beginning.