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AP Calc: LIM‑1 (EU), LIM‑1.C (LO), LIM‑1.C.1 (EK), LIM‑1.C.2 (EK), LIM‑1.C.3 (EK), LIM‑1.C.4 (EK)

So if we were to
ask ourselves, what is the value of our
function approaching-- as we approach x equals 2 from
values less than x equals 2. So as you imagine, as
we approach x equals 2-- So x equals 1, x equals 1.5,
x equals 1.9, x equals 1.999, x equals 1.99999999. What is f of x approaching? And we see that f of x
seems to be approaching this value right over here. It seems to be approaching 5. And so the way we
would denote that is the limit of f of x,
as x approaches 2-- and we're going to specify the
direction-- as x approaches 2 from the negative
direction-- we put the negative as a
superscript after the 2 to denote the direction
that we're approaching. This is not a negative 2. We're approaching 2 from
the negative direction. We're approaching 2
from values less than 2. We're getting closer and closer
to 2, but from below-- 1.9, 1.99, 1.99999 . As x gets closer and
closer from those values, what is f of x approaching? And we see here that
it is approaching 5. But what if we were asked the
natural other question-- What is the limit of f of
x as x approaches 2 from values greater than 2? So this is a little superscript
positive right over here. So now we're going to
approach x equals 2, but we're going to approach
it from this direction-- x equals 3, x equals 2.5, x
equals 2.1, x equals 2.01, x equals 2.0001. And we're going to get
closer and closer to 2, but we're coming from values
that are larger than 2. So here, when x equals
3, f of x is here. When x equals 2.5,
f of x is here. When x equals 2.01, f of x
looks like it's right over here. So in this situation, we're
getting closer and closer to f of x equaling 1. It never does quite equal that. It actually then just
has a jump discontinuity. This seems to be the limiting
value when we approach when we approach 2 from
values greater than 2. So this right over
here is equal to 1. And so when we think
about limits in general, the only way that a limit
at 2 will actually exist is if both of these
one-sided limits are actually the same thing. In this situation, they aren't. As we approach 2
from values below 2, the function seems
to be approaching 5. And as we approach 2
from values above 2, the function seems
to be approaching 1. So in this case, the limit--
let me write this down-- the limit of f of
x, as x approaches 2 from the negative
direction, does not equal the limit of f
of x, as x approaches 2 from the positive direction. And since this is the case--
that they're not equal-- the limit does not exist. The limit as x
approaches 2 in general of f of x-- so the limit of
f of x, as x approaches 2, does not exist. In order for it to have
existed, these two things would have had to have
been equal to each other. For example, if
someone were to say, what is the limit of f
of x as x approaches 4? Well, then we could think about
the two one-sided limits-- the one-sided limit from
below and the one-sided limit from above. So we could say,
well, let's see. The limit of f of x, as x
approaches 4 from below-- so let me draw that. So what we care
about-- x equals 4. As x equals 4 from below-- So when x equals 3,
we're here where f of 3 is negative 2. f of 3.5
seems to be right over here. f of 3.9 seems to be right
over here. f of 3.999-- we're getting closer and
closer to our function equaling negative 5. So the limit as we
approach 4 from below-- this one-sided
limit from the left, we could say-- this is going
to be equal to negative 5. And if we were to ask
ourselves the limit of f of x, as x approaches
4 from the right, from values larger than
4, well, same exercise. f of 5 gets us here. f of 4.5 seems
right around here. f of 4.1 seems right
about here. f of 4.01 seems right around here. And even f of 4 is
actually defined, but we're getting
closer and closer to it. And we see, once again,
we are approaching 5. Even if f of 4 was not
defined on either side, we would be
approaching negative 5. So this is also
approaching negative 5. And since the limit
from the left-hand side is equal to the limit
from the right-hand side, we can say-- so these
two things are equal. And because these
two things are equal, we know that the limit of
f of x, as x approaches 4, is equal to 5. Let's look at a
few more examples. So let's ask ourselves
the limit of f of x-- now, this is our new f of x depicted
here-- as x approaches 8. And let's approach
8 from the left. As x approaches 8 from
values less than 8. So what's this going
to be equal to? And I encourage you to pause the
video to try to figure it out yourself. So x is getting closer
and closer to 8. So if x is 7, f of 7 is here. If x is 7.5, f of 7.5 is here. So it looks like
our value of f of x is getting closer and
closer and closer to 3. So it looks like the limit
of f of x, as x approaches 8 from the negative
side, is equal to 3. What about from
the positive side? What about the
limit of f of x as x approaches 8 from the
positive direction or from the right side? Well, here we see as x
is 9, this is our f of x. As x is 8.5, this
is our f of 8.5. It seems like we're
approaching f of x equaling 1. So notice, these two
limits are different. So the non-one-sided limit,
or the two-sided limit, does not exist at f of
x or as we approach 8. So let me write that down. The limit of f of x,
as x approaches 8-- because these two things
are not the same value-- this does not exist. Let's do one more example. And here they're actually
asking us a question. The function f is graphed below. What appears to be the value of
the one-sided limit, the limit of f of x-- this is f of x--
as x approaches negative 2 from the negative direction? So this is the negative 2
from the negative direction. So we care what happens as
x approaches negative 2. We see f of x is actually
undefined right over there. But let's see what
happens as we approach from the negative
direction, or as we approach from values less
than negative 2, or as we approach from the left. As we approach from the
left, f of negative 4 is right over here. So this is f of negative 4. f of negative 3 is
right over here. f of negative 2.5 seems
to be right over here. We seem to be getting
closer and closer to f of x being equal
to 4, at least visually. So I would say that
it looks-- at least, graphically-- the limit of
f of x, as x approaches 2 from the negative
direction, is equal to 4. Now, if we also asked
ourselves the limit of f of x, as x
approaches negative 2 from the positive direction,
we would get a similar result. Now, we're going to
approach from when x is 0, f of x seems
to be right over here. When x is 1, f of x
is right over here. When x is negative
1, f of x is there. When x is negative 1.9, f of
x seems to be right over here. So once again, we seem to be
getting closer and closer to 4. Because the left-handed limit
and the right-handed limit are the same value. Because both one-sided limits
are approaching the same thing, we can say that the limit
of f of x, as x approaches negative 2-- and this
is from both directions. Since from both directions, we
get the same limiting value, we can say that the
limit exists there. And it is equal to 4.

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