[MUSIC] Up to now, we've been thinking a 
lot about limits. 
We've been considering f(x) when x is 
close to but not equal to a. 
Let's refine that a bit now. 
Instead of thinking about x close to a. 
I'm going to think about x close to the 
right side of a or x close to the left 
side of a. 
Super-sized definition. 
So we say the limit of f(x) as x 
approaches a from the right, I'm going to 
use this little + sign to mean from the 
right. 
So if this is equal to l, it means that 
f(x) is as close as you want to l, this 
is just like before, provided x is near 
enough a on the right-hand side. 
So we're going to use a little + sign to 
denote approaching from the right-hand 
side. 
Now we're going to play a same game for 
approaching from the left-hand side. 
So let's say the limit of f(x) as x 
approaches a from the left-hand side is 
equal to l means that f(x) is as close as 
you want to l just like usual, provided x 
is near enough a on the left-hand side. 
Let's go see some picture of this at the 
blackboard. 
Really helps to see a graph. 
Here's a graph of a made up function that 
I'm called f(x). 
You'll note that f(x) has some issues at 
the input 3. 
There's an empty circle here and a 
filled-in circle here so if I plug in 3, 
my output valley is 2. 
And indeed if I plug in numbers that are 
just a little bit above 3, I get out 
numbers that are close to 2. 
On the other hand, if I plug numbers that 
are a little bit less than 3, I get out 
numbers that are close to 1. 
We can summarize our observations with 
these two statements. 
The first is that the limit of f(x) as x 
approaches three from the right-hand side 
is equal to 2. 
And that makes sense because I can get 
the output of the function to be as close 
to 2 as I'd like if I'm willing to 
evaluate the function of inputs that are 
close to but just a little bit bigger 
than 3. 
Similarly, the limit of f(x) as x 
approaches two from the left-hand side is 
equal to 1. 
We think back in the graph, I was getting 
outputs that are close to one if I 
evaluated the function inputs which are 
close to, but just a little bit less, 
than 3. 
So in this case, the two sided limit 
doesn't exist. 
I can't say that f(x) is getting close to 
anything if all I know is that x is close 
to 3 but the left and the right hand 
limits do exist. I can say that f(x) is 
getting close to 1. If x approaches 3 
from the left-hand side, and I can say 
that f(x) is getting close to 2 if x 
approaches 3 from the right-hand side. 
This situation comes up quite a bit where 
you compute and you find that the right 
and the left hand limits exist but they 
disagree and consequently, the two side 
of limit doesn't exist. 
I can summarize it like this. 
If the limit of f(x) as x approaches a 
from the right, is different than the 
limit of f(x) as x approaches a from the 
left, then the two sided limit the limit 
of f(x) as x approaches a does not exist. 
It works the other way as well. 
If the limit from the right-hand side is 
equal to the limit from the left-hand 
side, let's call our common value l, then 
the two sided limit, the limit of f(x) as 
x approaches a, no + or -. 
So this is just the usual old limit. 
Then this limit exists and it's equal to 
that same common value, l. 
The homework will challenge you with many 
more situations of one sided limits. 
If you get stuck, contact us. 
We are here to help you succeed. 
[MUSIC]