AP®︎ Calculus AB (2017 edition)
- Types of discontinuities
- Classify discontinuities
- Worked example: point where a function is continuous
- Worked example: point where a function isn't continuous
- Limits of piecewise functions
- Removing discontinuities (factoring)
- Removing discontinuities (rationalization)
- Removable discontinuities
A function being continuous at a point means that the two-sided limit at that point exists and is equal to the function's value. Point/removable discontinuity is when the two-sided limit exists, but isn't equal to the function's value. Jump discontinuity is when the two-sided limit doesn't exist because the one-sided limits aren't equal. Asymptotic/infinite discontinuity is when the two-sided limit doesn't exist because it's unbounded.
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- Is an asymptotic discontinuity any different than an infinite discontinuity?(24 votes)
- They are the same thing – if you look in the
Abouttext it actually says "Asymptotic/infinite discontinuity".(23 votes)
- I understand that classification of discontinuities is 3 types
i. point removable
ii. jump discontinuities
anything also(4 votes)
- There are also oscillating discontinuities. Look at the graph of f(x)=sin(1/x). It has no value or limit at x=0.(5 votes)
- Hi, I am learning how to evaluate functions by direct substitution right now. I was wondering why simply substituting or re-arranging a function would automatically give us the limit at that point. For example, in the two graphs on the left in this video, the y-value is defined at the x-value but the limit either doesn't equal that same y-value or doesn't exist. I want to see the actual functions that could result in these two graphs to better understand why we can directly substitute without fear of scenarios like these two. Is it only possible for piece-wise functions to create these types of scenarios? I feel like I am overlooking something and would really appreciate the help. Thanks in advance!(4 votes)
- A function can be determined by direct substitution if and only if lim_(x->c)_ f(x) = f(c). In other words, as long as the function is not discontinuous, you can find the limit by direct substitution.
There is also another way to find the limit at another point, and that is by looking for a determinant for the indeterminate form by using other methods and defining it by using another function. For example, lim_(x->2) (x^2 + 4 x - 12)/(x - 2), determined directly, equals (0/0), indeterminant form. However, there are many ways to determine a function by simply simplifying the function when direct substitution yields the indeterminant form. For this example, you could simply factor the limit to get lim_(x->2)_ (x+6), x ≠ 2. The constraint is added to be mathematically correct when it comes to being equivalent to the limit beforehand. However, say you found a function that is similar to the simplified function, only without the constraint, called g(x) = (x+6). You can define that as your new limit: lim_(x->2) g(x) = 8, thus lim_(x->2) (x^2 + 4 x - 12)/(x - 2) = 8.
If you want to learn more, go to this page to see some more situations in which it's possible to do a direct substitution: https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-7/a/limit-strategies-flow-chart(1 vote)
- Is there any difference between saying that a limit doesn't exist and saying that a limit is unbounded? I've only ever heard Sal saying a limit doesn't exist/there is no limit when a limit is being taken from both sides. This is the first time I can remember Sal saying something similar regarding a limit being taken from one side, however he called the limit unbounded this time.
I hope I explained myself clearly and if anyone could answer this question, that would be appreciated, thanks!(3 votes)
- Is a quadractic formula discontinous(1 vote)
- Well, the quadratic formula is a formula, so it can't be graphed.
However, a function related to the quadratic formula, a quadratic polynomial, is continuous over its entire interval.(4 votes)
- I can't understand why the value of the y=x^2 graph at x=3 is 4, and not 9. Probably an obvious answer, but it's eluding me!(2 votes)
- What about the function which has one sided limit? For example if lim_(x->0) when approaching from the right exists, but lim_(x->0) when approaching from the left is asymptote ?
Does this qualify as asymptotic discontinuity or some kind of mix of jump/asymptotic discontinuity?(2 votes)
- That would be an asymptotic discontinuity. Asymptotic discontinuities are defined as occurring when at least one of the one-sided limits are undefined.(2 votes)
- Do sharp peaks or turns in the line, such as the ones in some parametric equations, count as a discontinuity?(1 vote)
- The derivative of said functions would be discontinuous, but as long as the line never breaks it is continuous. Still, sharp turns or other sudden changes in slope will make the function non differentiable. So still something you have to keep an eye out for.(4 votes)
- What would be an example of a function with both a removable and a non-removable discontinuity?(2 votes)
- [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.