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# 1-sided vs. 2-sided limits (more examples)

Video transcript

So we have a function, f of
x, graphed right over here. And then we have a
bunch of statements about the limit of f of x, as
x approaches different values. And what I want to do is figure
out which of these statements are true and which
of these are false. So let's look at
this first statement. Limit of f of x, as x approaches
1 from the positive direction, is equal to 0. So is this true or false? So let's look at it. So we're talking about
as x approaches 1 from the positive direction,
so for values greater than 1. So as x approaches 1 from
the positive direction, what is f of x? Well, when x is, let's say 1
and 1/2, f of x is up here, as x gets closer and closer
to 1, f of x stays right at 1. So as x approaches 1 from
the positive direction, it looks like the limit of
f of x as x approaches 1 from the positive
direction isn't 0. It looks like it is 1. So this is not true. This would be true
if instead of saying from the positive
direction, we said from the negative direction. From the negative direction,
the value of the function really does look like
it is approaching 0. For approaching 1 from the
negative direction, when x is right over
here, this is f of x. When x is right over
here, this is f of x. When x is right over
here, this is f of x. And we see that
the value of f of x seems to get closer
and closer to 0. So this would only
be true if they were approaching from
the negative direction. Next question. Limit of f of x, as x approaches
0 from the negative direction, is the same as limit of
f of x as x approaches 0 from the positive direction. Is this statement true? Well, let's look. Our function, f of
x, as we approach 0 from the negative
direction-- I'm using a new color--
as we approach 0 from the negative direction,
so right over here, this is our value of f of x. Then as we get closer, this
is our value of f of x. As we get even closer, this
is our value of f of x. So it seems from the
negative direction like it is approaching
positive 1. From the positive direction,
when x is greater than 0, let's try it out. So if, say, x is 1/2,
this is our f of x. If x is, let's say,
1/4, this is our f of x. If x is just barely larger
than 0, this is our f of x. So it also seems
to be approaching f of x is equal to 1. So this looks true. They both seem to be
approaching the limit of 1. The limit here is 1. So this is absolutely true. Now let's look at
this statement. The limit of f of
x, as x approaches 0 from the negative
direction, is equal to 1. Well, we've already
thought about that. The limit of f of
x, as x approaches 0 from the negative
direction, we see that we're getting
closer and closer to 1. As x gets closer and
closer to 0, f of x gets closer and closer to 1. So this is also true. Limit of f of x, as x
approaches 0 exists. Well, it definitely exists. We've already established
that it's equal to 1. So that's true. Now the limit of f of x
as x approaches 1 exists, is that true? Well, we already saw that
when we were approaching 1 from the positive
direction, the limit seems to be approaching 1. We get when x is 1
and 1/2, f of x is 1. When x is a little bit
more than 1, it's 1. So it seems like we're getting
closer and closer to 1. So let me write that down. The limit of f of
x, as x approaches 1 from the positive
direction, is equal to 1. And now what's the limit
of f of x as x approaches 1 from the negative direction? Well, here, this is our f of x. Here, this is our f of x. It seems like our f of x is
getting closer and closer to 0, when we approach 1 from
values less than 1. So over here it equals 0. So if the limit from
the right-hand side is a different value than the
limit from the left-hand side, then the limit does not exist. So this is not true. Now finally, the limit of f
of x, as x approaches 1.5, is equal to 1. So right over here. So everything we've been
dealing with so far, we've always looked at
points of discontinuity, or points where maybe the
function isn't quite defined. But here, this is kind
of a plain vanilla point. When x is equal to 1.5,
that's maybe right over here, this is f of 1.5. That right over
there is the point, well, this is the
value f of 1.5. We could say f of, we could see
that f of 1.5 is equal to 1, that this right here is
the point 1.5 comma 1. And if we approach it
from the left-hand side, from values less than it, it's
1, the limit seems to be 1. When we approach from
the right-hand side, the limit seems to be 1. So this is a pretty
straightforward thing. The graph is continuous
right there, and so really, if we just substitute
at that point, or we just look at
the graph, the limit is the value of
the function there. You don't have to have a
function be undefined in order to find a limit there. So it is, indeed, the
case that the limit of f of x, as x approaches
1.5, is equal to 1.