[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]