Welcome back.
Now that we hopefully have a little bit of an intuition of
what a limit is, or finding the limit of a function is,
let's do some problems.
Some of these you might actually see on your exams or
when you're actually trying to solve a general limit problem.
So let's say what is the limit-- once again, my
pen is not working.
What is the limit as x approaches-- let's
say negative 1.
And let me see, what's a good-- let's say my expression is--
I'll put it in parentheses so it's cleaner.
It's 2x plus 2 over x plus 1.
So the first thing I would always try to do is just say
what happens if I just stick x straight into this expression?
What happens?
Well, what's 2x plus 2 when x is equal to negative 1?
2 times negative 1.
2 times negative 1 plus 2 over negative 1 plus 1.
Well, the numerator is negative 2 plus 2-- that equals 0--
over-- what's the denominator?
Negative 1 plus 1.
Over 0.
And do we know what 0 over 0 is?
Well, no.
It's undefined, right?
So here's a case, just like what we saw in that first
video, where the limit actually can't equal what the expression
equals when you substitute x for the number you're trying to
find the limit of because you get an undefined answer.
So let's see if, using the limit, we can come up
with a better answer for what it's approaching.
Well, since we're just starting with these limit problems,
let me draw a graph.
And I think this is going to give you the intuition
for what we're doing.
It'll probably give you the answer.
But then I'll show you how to solve this analytical.
So if I draw a graph, these are the axes.
Actually, I'll do the graphical and the analytical
at the same time.
So I want to rewrite this expression in a way that
maybe I can simplify it.
So 2x plus 2.
Isn't that the same thing as 2 times x plus 1?
2 times x plus 1, right?
2x plus 2 is the same thing as 2 times x plus one, and then
all of that is over x plus 1.
So as long as this expression and this expression don't equal
0, it actually turns out that this function-- let's say
this is f of x, right?
This function.
Well, for every value other than x is equal to negative
1, you could actually cancel this and this out.
And so really, we see that f of x is equal to-- I need to find
a better tool-- f of x is equal to 2 when x does not
equal negative 1.
And we saw when x is equal to negative 1, it's undefined.
So undefined when equals negative 1.
So how would we graph that?
We showed that f of x is equal to 2 when x does not equal
negative 1 and f of x is undefined when x
equals negative 1.
And once again, all I did is kind of rewrite this exact
same function, right?
I showed that I could simplify and I could divide the
numerator and denominator by x plus 1 as long as x does not
equal negative 1, and that otherwise, it's undefined.
So let me graph this.
I'm going to get a different color.
Maybe I'll go with red.
So this is 2.
So we see that x is-- and let me say this is negative 1.
So for every other value other than negative 1, the value of
this, of f of x, is equal to 2.
This is 1, this is 2, this is 3, and so on.
At negative 1, the graph is undefined.
So there's a hole there.
And then we keep going to the left-hand side.
So if we're going to do the limit, we can just visually
say, well, as x-- let me do another color now.
As x comes from the left-hand side, what does f of x equal?
Well, f of x is 2, 2, 2, 2, 2. f of x is equal to 2
until we get to exactly negative 1, right?
And similarly, when we go from the other hand, it's
the exact same thing.
f of x is 2, 2, 2 until we get to negative 1.
So you'll see, and I'll make sure you see it visually here,
that the limit as approaches negative 1 of 2x plus 2 over
x plus 1, it equals 2.
Let me draw a line here so you don't get messed
up with all of it.
And I'm not formally, I guess, proving here that the limit is
2, but I'm showing you kind of an analytical way, and this is
actually how it tends to be done in algebra class, is that
you tend to simplify the expression so that you say, oh,
if there wasn't a hole here, what would the f of
x equal, right?
And then you'd just evaluate it at that point.
I think this might give you a little intuition, but this
isn't a formal solution.
But unless you're asked to, you tend not to be asked
for a formal solution.
You actually just tend to ask what the limit is, and this is
the way you could solve it.
And actually another way that you could-- I mean, I often
used to check my answers when I used to do it is you could take
a calculator and try in-- what happens when: what is f
of minus 1.001, right?
And you can also try what is f of negative 0.999, right?
Because what you want to do is you want to say, well, what
does the function equal when I get really close to negative 1?
And then you could keep going closer and closer to negative
1 and see what the function approaches, and in this
case, you'll see that it approaches 2.
So let's do another problem.
Well, let's say, what is the limit as x approaches
0 of 1 over x?
I think here it might be useful to draw this graph because
it'll give you a visual reason, a visual represent-- actually,
let's do it both ways.
Let's say-- let's do it the picking-numbers method because
I think that'll give you an intuition and maybe it'll
help us draw the graph.
So let's say that this is f of x.
f of x-- you can tell my presentation is very
unplanned-- f of x is equal to 1 over x.
And we want to find the limit as x approaches 0.
So what is f of-- actually, let's make a table.
f of x.
So clearly when x is equal to 0, we don't know.
It's undefined.
1 over 0 is undefined.
But what happens when x equals minus 0.01?
Well, with minus 0.01, 1 over minus 0.01, that is equal
to negative 100, right?
What happens when x is equal to minus 0.001, right?
So we're getting closer and closer to 0 from
the negative direction.
Well, here it equals-- make sure my pen is
working, color right.
Something's wrong with my tool.
Now my computer's breaking down.
Let's see what's going on.
I think my computer just froze.
Well, I'm going to try to solve this, and in the very next
video, I'm going to continue on with this problem.
So I'll actually see you in the next presentation while I
figure out why my pen isn't working, and then we'll
continue with this problem.
See you very soon.