Hi guys! I'm Nancy.
And this is my introduction to limits.
I will show you what a limit is.. how to read its notation..
and what it means on a graph.
So if you actually just wanted to know
how to find the limit from the limit expression, the x expression..
I have videos for those.
"How to Find the Limit" and "How to Find the Limit at Infinity".
And you can use the links in the video or in the description to jump to those.
But here, let me introduce you to limits.
So what is a limit?
It's basically just a prediction of what y-value it looks like we should get at a point.
That seems kinda vague, so let me show you what I mean.
Look at this graph and this limit question
The graph's a little strange, it has a hole in it, a gap, but that's completely possible.
What does this limit notation mean, when you see this?
Well, officially, technically how you read this is
"the limit, as x approaches 1, of f(x) equals" some number.
What does that mean? That means..
When x gets really close to 1, what is y getting really close to?
And the limit, the answer to the limit, is always a y-value.
So in other words, as x tends toward 1, or heads towards 1
what is y tending toward, or heading toward?
Another way of thinking of it is.. if we put in..
input x's really close to 1.. what kind of y's are we getting out? As output.
Why do we even need the limit by the way?
One reason is cases like this
 where we can't "see" what's actually happening exactly at a point.
We can only just see what's happening around that point.
So how do you find this limit?
There's a hidden meaning here..
when you see something like this, the limit as x approaches 1
it actually means you have to check both sides of x = 1.
So it's not spelled out for you, it's not obvious
but there's this hidden implied meaning that you check
from the left of 1, and from the right of 1, in order to find the limit.
How do you check from the left of 1?
Well, on a graph it means
you'll trace x-values heading toward x = 1 from the left.
From the left side, so we're actually heading right.
But from the left side, towards x = 1.
But on the graph. Wherever that may be on the function.
So if we trace from the left, heading towards x = 1
what we really care about for the limit is y-values.
The vertical values.
So we need to check to see what y-values we're hitting
as we approach from the left.
And it will be numbers like 1.5.. 1.6.. 1.7..
Getting really close to 1.9.
And if we get incredibly close here and zoom in
we'll see numbers like 1.99.. 1.999999..
Really close to 2. Practically 2.
So the limit from the left is 2.
We also have to check from the right.
How do you do that?
You trace x-values from the right headed towards x = 1.
But on the graph. On the function.
So in that case, heading from the right
in this graph, kinda looks like heading down
but from the right, you check to see what y-values you're hitting.
And it will be numbers like 2.4.. 2.3.. 2.2.. 2.1.
And if you get really really close you'll see like.. 2.01.. 2.0001..
Basically 2. From the right, it's heading toward 2.
And the limit is 2, from the right.
This is great because
if the limit from the right is the same as the limit from the left..
2 and 2 for the y we're hitting..
then we can say that the overall limit is also 2.
We can write here that the limit..
is equal to 2.
Remember, that limit answer is always a y-value.
There might have been some x-value here..
but the limit always a y-value.
And one thing that confuses people is the equal sign..
Some people wonder, you know, why can you use an equal sign?
How can you say it's equal if you were just saying that it's approaching a number
or getting really close and not actually touching exactly the number?
That's a great question actually.
It's just the limit notation. It's just the limit language.
We can say that the limit equals a number
and it just means that that number is what the function is approaching.
And notice that I'm not saying that the function equals 2 at x = 1.
It's not that.
And in fact, if you look at the graph, there's a hole at x = 1.
We don't even know what the function equals.
It's not defined. It's indeterminate.
Even though there's this hole at that point.. x = 1
We can still say that the limit, as x approaches 1, is equal to 2.
OK. Here's where it gets even stranger.
Right now there's a hole here
and this function is not defined there.
What if I define it? And put a point above or below the hole?
That looks pretty weird.
But you'll probably see one like this.
This is meant to test your understanding of the limit.
Even though there's this point here where the function is defined at 1.
It's defined to be 3..
Turns out for the limit
we don't even care what the function is doing exactly at 1.
Even though there's this point up here.
For the limit, we only care what's happening just around 1
and not actually exactly at 1.
That's important.
So since, even though the point is there..
Since it's still approaching a y of 2 from the left and from the right
the limit is still 2.
OK. What if you have a very normal looking function like this..
Just a line.
And you still want to find the limit, as x approaches 1, of f(x).
Well, it's the same exact idea. The same thing that you do.
You see what's happening from the left and from the right.
We don't care what's actually happening exactly at that point x =1.
It's nice that it's a normal continuous line.
And that it happens to equal 2 exactly at that point.
But we don't care. We shouldn't care.
We shouldn't even look at that for the limit.
For the limit, strictly
all we should check is what y it's approaching from the left and from the right.
Seems weird in this case.. it's such a normal looking function.
But you could get one like this.
To test your understanding of the limit.
And since it's approaching 2 from the left and from the right
the limit is still 2.
OK. So sometimes the left side limit and right side limit are not the same.
Take a look at this graph. Looks kinda like a step function.
It jumps suddenly here.. from 3 to 4.
Say that you want to find the limit, as x approaches 2, of f(x)..
How do you do that?
It's the same idea.
You will have to check from both sides.
From the left side and the right side.
From here.. and from here..
in order to give an answer for that overall limit.
It's a really good chance for me to show you one-sided limits.
The left-sided limit and the right-sided limit.
If you see this notation
This means the limit as x approaches 2 from the left.. the minus sign means.
And the plus sign here means the limit as x approaches 2 from the right.
So we do have to go through those anyway to find the answer.
So I wrote them out here for you.
Let's find the limit as x approaches 2 from the left.
It's just like before, you wanna trace from the left toward 2.
Toward 2, but on the graph.
And as you do that, you look for what y-values it's hitting.
And since this is just a flat horizontal line..
everywhere along here it is 3. For the y.
So the limit from the left is 3.
And then from the right.. from the right side..
You approach from the right.
Approaching 2 from the right, but on the graph, on the function.
And if you do that, everywhere here it's 4 for the y-value.
So the limit, even if you get really close..
Looks like it's headed toward 4.
So if somebody asked you just for the left-sided limit
or just for the right-sided limit
You would be done. The left is 3. The right is 4.
But if someone's asking you for the overall one, you have to compare them.
And since they're not the same.. 3 is not equal to 4..
We can't give a number for the overall limit.
It does not exist.
So we could write "DNE" or does not exist.
So basically even though the overall limit did not exist
there were these left-sided and right-sided limits
that we could write values for.
OK. One last kind, I promise!
What if the limit is as x is going off to infinity or negative infinity?
This is another kind of strange blind spot case
because we can never actually see at infinity or negative infinity.
We can only approach it and see what's happening to the y-value.
So how would you find this? The limit, as x approaches infinity, of f(x).
If x needs to approach infinity
it needs to approach a very large positive number
to head in that direction.
So we want to go toward the right when we're tracing the graph.
The function.
So on your function, you want to trace toward the right..
off to the right..
and see what's happening to the y-value.
You can kind of tell that this function is leveling off
toward this asymptote that is at y = 0.
It's heading toward it. It will never reach it.
But it looks like it's getting very very close to 0
the farther we get toward infinity.
So the limit would be 0.
What about for the limit as x approaches negative infinity?
We can never reach it, but we can move in that direction
which is the left direction.. the very large negative direction..
on the function.
So as we trace in that direction
we want to pay attention to what's happening to the y-value.
It looks like it's reaching this -1 asymptote.
Getting closer and closer..
So the limit will be -1. That y-value.
So for limits at infinity
we can't actually reach those values: infinity and negative infinity
but we can see what's happening and give y-values for the limits.
So I hope that helped you get started with limits.
If it did, please 'Like' my video!
Unfortunately that was just the tip of the limits iceberg.
So here are some links to my other videos
to help you find specific kinds of limits.
And of course you can always Subscribe to my channel
which will make it easier to find if you're pulling an all-nighter.
Just sayin'..
And I wish you the best of luck with limits!
