If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:7:16

AP Calc: LIM‑2 (EU), LIM‑2.A (LO), LIM‑2.A.1 (EK)

- [Instructor] What we're
going to do in this video is talk about the various
types of discontinuities that you've probably seen
when you took algebra, or precalculus, but then
relate it to our understanding of both two-sided limits
and one-sided limits. So let's first review the
classification of discontinuities. So here on the left,
you see that this curve looks just like y equals x squared, until we get to x equals three. And instead of it being three squared, at this point you have this opening, and instead the function at
three is defined at four. But then it keeps going
and it looks just like y equals x squared. This is known as a point, or a removable, discontinuity. And it's called that for obvious reasons. You're discontinuous at that point. You might imagine defining
or redefining the function at that point so it is continuous, so that this discontinuity is removable. But then how does this
relate to our definition of continuity? Well, let's remind ourselves
our definition of continuity. We say f is continuous, continuous, if and only if, or let me write f continuous at x equals c, if and only if the limit as x approaches c of f of x is equal to the
actual value of the function when x is equal to c. So why does this one fail? Well, the two-sided limit actually exists. You could find, if we say
c in this case is three, the limit as x approaches three of f of x, it looks like, and if you
graphically inspect this, and I actually know this is the
graph of y equals x squared, except at that discontinuity
right over there, this is equal to nine. But the issue is, the way
this graph has been depicted, this is not the same thing
as the value of the function. This function f of three, the way it's been graphed, f of three is equal to four. So this is a situation where
this two-sided limit exists, but it's not equal to the
value of that function. You might see other
circumstances where the function isn't even defined there, so that isn't even there. And so, once again, the limit might exist, but the function might
not be defined there. So, in either case, you aren't
going to meet this criteria for continuity. And so that's how a point
or removable discontinuity, why it is discontinuous with regards to our limit
definition of continuity. So now let's look at this second example. If we looked at our
intuitive continuity test, if we would just try to trace this thing, we see that once we get to x equals two, I have to pick up my
pencil to keep tracing it. And so that's a pretty good
sign that we are discontinuous. We see that over here as well. If I'm tracing this function,
I gotta pick up my pencil to, I can't go to that point. I have to jump down here, and then keep going right over there. So in either case I have
to pick up my pencil. And so, intuitively, it is discontinuous. But this particular type of discontinuity, where I am making a jump from one point, and then I'm making a jump
down here to continue, it is intuitively called a jump discontinuity, discontinuity. And this is, of course, a
point removable discontinuity. And so how does this relate to limits? Well, here, the left and
right-handed limits exist, but they're not the same thing, so you don't have a two-sided limit. So, for example, for
this one in particular, for all the x-values up to
and including x equals two, this is the graph of y equals x squared. And then for x greater than two, it's the graph of square root of x. So in this scenario, if you were to take the limit of f of x as x approaches two from the left, from the left, this is going to be equal to four, you're approaching this value. And that actually is the
value of the function. But if you were to take the
limit as x approaches two from the right of f of x, what is that going to be equal to? Well, approaching from the right, this is actually the square root of x, so it's approaching
the square root of two. You wouldn't know it's
the square root of two just by looking at this. I know that, just because when I, when I went on to Desmos
and defined the function, that's the function that I used. But it's clear even visually that you're approaching
two different values when you approach from the left than when you approach from the right. So even though the one-sided limits exist, they're not approaching the same thing, so the two-sided limit doesn't exist. And if the two-sided limit doesn't exist, it for sure cannot be equal to the value of the function there, even
if the function is defined. So that's why the jump
discontinuity is failing this test. Now, once again, it's intuitive. You're seeing that, hey, I gotta jump, I gotta pick up my pencil. These two things are not
connected to each other. Finally, what you see here is, when you learned precalculus, often known as an
asymptotic discontinuity, asymptotic, asymptotic discontinuity, discontinuity. And, intuitively, you
have an asymptote here. It's a vertical asymptote at x equals two. If I were to try to trace the graph from the left, I would just keep on going. In fact, I would be doing
it forever, 'cause it's, it would be infinitely, it would be unbounded as
I get closer and closer to x equals two from the left. And if try to get to x
equals two from the right, once again I get unbounded up. But even if I could, and when I say it's unbounded,
it goes to infinity, so it's actually impossible in a mortal's lifespan to
try to trace the whole thing. But you get the sense that,
hey, there's no way that I could draw from here to here
without picking up my pencil. And if you wanna relate it
to our notion of limits, it's that both the left and right-handed
limits are unbounded, so they officially don't exist. So if they don't exist, then
we can't meet these conditions. So if I were to say, the limit as x approaches two from the
left-hand side of f of x, we can see that it goes unbounded
in the negative direction. You might sometimes see someone
write something like this, negative infinity. But that's a little
handwavy with the math. The more correct way to say
it is it's just unbounded, unbounded. And, likewise, if we
thought about the limit as x approaches two from the right of f of x, it is now unbounded
towards positive infinity. So this, once again, this is also, this is also unbounded. And because it's unbounded and
this limit does not exist, it can't meet these conditions. And so we are going to be discontinuous. So this is a point or
removable discontinuity, jump discontinuity, I'm jumping, and then we have these
asymptotes, a vertical asymptote. This is an asymptotic discontinuity.