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Limits intro

In this video, we learn about limits, a fundamental concept in calculus. Limits help us understand what a function approaches as the input gets closer to a certain value, even when the function is undefined at that point. The video demonstrates this concept using two examples with different functions. Created by Sal Khan.

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  • blobby green style avatar for user Niharika Tiwari
    Do one-sided limits count as a real limit or is it just a concept that is really never applied?
    (238 votes)
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    • leaf blue style avatar for user Matthew Daly
      It does get applied in finding real limits sometimes, but it is not usually a "real limit" itself. For instance, let f be the function such that f(x) is x rounded to the nearest integer. What is the limit of f(x) as x approaches 0.5? Well, there isn't one, and the reason is that even though the left-hand limit and the right-hand limit both exist, they aren't equal to each other.

      Of course, if a function is defined on an interval and you're trying to find the limit of the function as the value approaches one endpoint of the interval, then the only thing that makes sense is the one-sided limit, since the function isn't defined "on the other side".
      (252 votes)
  • blobby green style avatar for user Aliza Lazarus
    Does anyone know where i can find out about practical uses for calculus?
    (78 votes)
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    • leaf green style avatar for user andrewpolidori
      The amount of practical uses for calculus are incredibly numerous, it features in many different aspects of life from Finance to Life Sciences to Engineering to Physics. It's hard to point to a place where you could go to find out about the practical uses of calculus, because you could go almost anywhere. I recommend doing a quick Google search and you'll find limitless (pardon the pun) examples.
      (310 votes)
  • leaf green style avatar for user Ismail
    What exactly is definition of Limit?
    (29 votes)
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    • blobby green style avatar for user Charles
      The strictest definition of a limit is as follows: Say Aₓ is a series. If there exists a real number L that for any positive value Ԑ (epsilon), no matter how small, there exists a natural number X, such that { |Aₓ - L| < Ԑ, as long as x > X }, then we say A is limited by L, or L is the limit of A, written as lim (x→∞) A = L.

      This is usually what is called the Ԑ - N definition of a limit. (I replaced the n's and N's in the equations with x's and X's, because I couldn't find a symbol for subscript n).
      (21 votes)
  • aqualine sapling style avatar for user shaylagale24
    Are there any textbooks that go along with these lessons? It would be great to have some exercises to go along with the videos.
    (9 votes)
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  • blobby green style avatar for user Junie
    thank god for khan academy bc my college calc teacher is functionally useless
    (28 votes)
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  • leaf orange style avatar for user Rebecca Gray
    0/0 seems like it should equal 0. Why doesn't it?
    (0 votes)
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  • blobby green style avatar for user Veronica Batallones
    Would that mean, if you had the answer 2/0 that would come out as undefined right? since x/0 is undefined :( just want to clarify
    (6 votes)
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  • blobby green style avatar for user Abhishek Abi
    What is the difference between calculus and other forms of maths like arithmetic, geometry, algebra, i.e., what special about calculus over these(i see lot of basic maths are used in calculus, are these structured in our school level maths to learn calculus!!).
    (8 votes)
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    • leaf grey style avatar for user Qeeko
      Elementary calculus may be described as a study of real-valued functions on the real line. One divides these functions into different classes depending on their properties. Examples of such classes are the continuous functions, the differentiable functions, the integrable functions, etc.

      Some insight will reveal that this process of grouping functions into classes is an attempt to categorize functions with respect to how "smooth" or "well-behaved" they are. For instance, an integrable function may be less smooth (in some appropriate sense) than a continuous function, which may be less smooth than a differentiable function, which may be less smooth than a twice differentiable function, and so on.

      If one knows that a function ƒ is continuous, what else can you say about ƒ? The intermediate value theorem, the extreme value theorem, and so on, are examples of theorems describing further properties enjoyed by continuous functions. One should regard these theorems as descriptions of the various classes.

      And then there is, of course, the computational aspect. Elementary calculus is also largely concerned with such questions as how does one compute the derivative of a differentiable function? How does one compute the integral of an integrable function? Here there are many techniques to be mastered, e.g., the product rule, the chain rule, integration by parts, change of variable in an integral.

      Some calculus courses focus most on the computational aspects, some more on the theoretical aspects, and others tend to focus on both.

      The reason you see a lot of, say, algebra in calculus, is because many of the definitions in the subject are based on the algebraic structure of the real line. Many aspects of calculus also have geometric interpretations in terms of areas, slopes, tangent lines, etc.
      (15 votes)
  • leaf grey style avatar for user Erik Jackson
    I'm not quite sure I understand the full nature of the limit, or at least how taking the limit is any different than solving for Y. I understand that if a function is undefined at say, 3, that it cannot be solved at 3. However, wouldn't taking the limit as X approaches 3.00001 or 2.99999 be the same as solving for X at these points? I'm sure I'm missing something.
    (7 votes)
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    • piceratops ultimate style avatar for user Just Keith
      A limit is a method of determining what it looks like the function "ought to be" at a particular point based on what the function is doing as you get close to that point. If you have a continuous function, then this limit will be the same thing as the actual value of the function at that point. In fact, that is one way of defining a continuous function:
      A continuous function is one where
      f(c) = lim x→c⁻ f(x) = lim x→c⁺ f(x) for all values of c within the domain.

      But, suppose that there is something unusual that happens with the function at a particular point. If the function is not continuous, even if it is defined, at a particular point, then the limit will not necessarily be the same value as the actual function.
      Suppose we have the function:
      f(x) = 2x, where x≠3,
      and 200, where x=3

      So, this function has a discontinuity at x=3.
      Thus, f(3) = 200
      But lim x→3 f(x) = 6, because, it looks like the function ought to be 6 when you get close to x=3, even though the actual function is different.

      The other thing limits are good for is finding values where it is impossible to actually calculate the real function's value -- very often involving what happens when x is ±∞. Since ∞ is not a number, you cannot plug it in and solve the problem. But you can use limits to see what the function ought be be if you could do that.
      lim x→+∞ (2x² + 5555x +2450) / (3x²)
      We can determine this limit by seeing what f(x) equals as we get really large values of x.
      f(10) = 194
      f(10⁴) ≈ 0.8518
      f(10⁶) ≈ 0.6685185
      f(10¹⁰) ≈ 0.66666685
      f(10²⁰) ≈ 0.666666666666666685
      Quite clearly as x gets large and larger, this function is getting closer to ⅔, so the limit is ⅔.

      Note: using l'Hopital's Rule and other methods, we can exactly calculate limits such as these, so we don't have to go through the effort of checking like this.
      (11 votes)
  • starky seedling style avatar for user Debanik
    why it is important to check limit from both sides of a function?
    (6 votes)
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Video transcript

In this video, I want to familiarize you with the idea of a limit, which is a super important idea. It's really the idea that all of calculus is based upon. But despite being so super important, it's actually a really, really, really, really, really, really simple idea. So let me draw a function here, actually, let me define a function here, a kind of a simple function. So let's define f of x, let's say that f of x is going to be x minus 1 over x minus 1. And you might say, hey, Sal look, I have the same thing in the numerator and denominator. If I have something divided by itself, that would just be equal to 1. Can't I just simplify this to f of x equals 1? And I would say, well, you're almost true, the difference between f of x equals 1 and this thing right over here, is that this thing can never equal-- this thing is undefined when x is equal to 1. Because if you set, let me define it. Let me write it over here, if you have f of, sorry not f of 0, if you have f of 1, what happens. In the numerator, we get 1 minus 1, which is, let me just write it down, in the numerator, you get 0. And in the denominator, you get 1 minus 1, which is also 0. And so anything divided by 0, including 0 divided by 0, this is undefined. So you can make the simplification. You can say that this is you the same thing as f of x is equal to 1, but you would have to add the constraint that x cannot be equal to 1. Now this and this are equivalent, both of these are going to be equal to 1 for all other X's other than one, but at x equals 1, it becomes undefined. This is undefined and this one's undefined. So how would I graph this function. So let me graph it. So that, is my y is equal to f of x axis, y is equal to f of x axis, and then this over here is my x-axis. And then let's say this is the point x is equal to 1. This over here would be x is equal to negative 1. This is y is equal to 1, right up there I could do negative 1. but that matter much relative to this function right over here. And let me graph it. So it's essentially for any x other than 1 f of x is going to be equal to 1. So it's going to be, look like this. It's going to look like this, except at 1. At 1 f of x is undefined. So I'm going to put a little bit of a gap right over here, the circle to signify that this function is not defined. We don't know what this function equals at 1. We never defined it. This definition of the function doesn't tell us what to do with 1. It's literally undefined, literally undefined when x is equal to 1. So this is the function right over here. And so once again, if someone were to ask you what is f of 1, you go, and let's say that even though this was a function definition, you'd go, OK x is equal to 1, oh wait there's a gap in my function over here. It is undefined. So let me write it again. It's kind of redundant, but I'll rewrite it f of 1 is undefined. But what if I were to ask you, what is the function approaching as x equals 1. And now this is starting to touch on the idea of a limit. So as x gets closer and closer to 1. So as we get closer and closer x is to 1, what is the function approaching. Well, this entire time, the function, what's a getting closer and closer to. On the left hand side, no matter how close you get to 1, as long as you're not at 1, you're actually at f of x is equal to 1. Over here from the right hand side, you get the same thing. So you could say, and we'll get more and more familiar with this idea as we do more examples, that the limit as x and L-I-M, short for limit, as x approaches 1 of f of x is equal to, as we get closer, we can get unbelievably, we can get infinitely close to 1, as long as we're not at 1. And our function is going to be equal to 1, it's getting closer and closer and closer to 1. It's actually at 1 the entire time. So in this case, we could say the limit as x approaches 1 of f of x is 1. So once again, it has very fancy notation, but it's just saying, look what is a function approaching as x gets closer and closer to 1. Let me do another example where we're dealing with a curve, just so that you have the general idea. So let's say that I have the function f of x, let me just for the sake of variety, let me call it g of x. Let's say that we have g of x is equal to, I could define it this way, we could define it as x squared, when x does not equal, I don't know when x does not equal 2. And let's say that when x equals 2 it is equal to 1. So once again, a kind of an interesting function that, as you'll see, is not fully continuous, it has a discontinuity. Let me graph it. So this is my y equals f of x axis, this is my x-axis right over here. Let me draw x equals 2, x, let's say this is x equals 1, this is x equals 2, this is negative 1, this is negative 2. And then let me draw, so everywhere except x equals 2, it's equal to x squared. So let me draw it like this. So it's going to be a parabola, looks something like this, let me draw a better version of the parabola. So it'll look something like this. Not the most beautifully drawn parabola in the history of drawing parabolas, but I think it'll give you the idea. I think you know what a parabola looks like, hopefully. It should be symmetric, let me redraw it because that's kind of ugly. And that's looking better. OK, all right, there you go. All right, now, this would be the graph of just x squared. But this can't be. It's not x squared when x is equal to 2. So once again, when x is equal to 2, we should have a little bit of a discontinuity here. So I'll draw a gap right over there, because when x equals 2 the function is equal to 1. When x is equal to 2, so let's say that, and I'm not doing them on the same scale, but let's say that. So this, on the graph of f of x is equal to x squared, this would be 4, this would be 2, this would be 1, this would be 3. So when x is equal to 2, our function is equal to 1. So this is a bit of a bizarre function, but we can define it this way. You can define a function however you like to define it. And so notice, it's just like the graph of f of x is equal to x squared, except when you get to 2, it has this gap, because you don't use the f of x is equal to x squared when x is equal to 2. You use f of x-- or I should say g of x-- you use g of x is equal to 1. Have I been saying f of x? I apologize for that. You use g of x is equal to 1. So then then at 2, just at 2, just exactly at 2, it drops down to 1. And then it keeps going along the function g of x is equal to, or I should say, along the function x squared. So my question to you. So there's a couple of things, if I were to just evaluate the function g of 2. Well, you'd look at this definition, OK, when x equals 2, I use this situation right over here. And it tells me, it's going to be equal to 1. Let me ask a more interesting question. Or perhaps a more interesting question. What is the limit as x approaches 2 of g of x. Once again, fancy notation, but it's asking something pretty, pretty, pretty simple. It's saying as x gets closer and closer to 2, as you get closer and closer, and this isn't a rigorous definition, we'll do that in future videos. As x gets closer and closer to 2, what is g of x approaching? So if you get to 1.9, and then 1.999, and then 1.999999, and then 1.9999999, what is g of x approaching. Or if you were to go from the positive direction. If you were to say 2.1, what's g of 2.1, what's g of 2.01, what's g of 2.001, what is that approaching as we get closer and closer to it. And you can see it visually just by drawing the graph. As g gets closer and closer to 2, and if we were to follow along the graph, we see that we are approaching 4. Even though that's not where the function is, the function drops down to 1. The limit of g of x as x approaches 2 is equal to 4. And you could even do this numerically using a calculator, and let me do that, because I think that will be interesting. So let me get the calculator out, let me get my trusty TI-85 out. So here is my calculator, and you could numerically say, OK, what's it going to approach as you approach x equals 2. So let's try 1.94, for x is equal to 1.9, you would use this top clause right over here. So you'd have 1.9 squared. And so you get 3.61, well what if you get even closer to 2, so 1.99, and once again, let me square that. Well now I'm at 3.96. What if I do 1.999, and I square that? I'm going to have 3.996. Notice I'm going closer, and closer, and closer to our point. And if I did, if I got really close, 1.9999999999 squared, what am I going to get to. It's not actually going to be exactly 4, this calculator just rounded things up, but going to get to a number really, really, really, really, really, really, really, really, really close to 4. And we can do something from the positive direction too. And it actually has to be the same number when we approach from the below what we're trying to approach, and above what we're trying to approach. So if we try to 2.1 squared, we get 4.4. If we do 2. let me go a couple of steps ahead, 2.01, so this is much closer to 2 now, squared. Now we are getting much closer to 4. So the closer we get to 2, the closer it seems like we're getting to 4. So once again, that's a numeric way of saying that the limit, as x approaches 2 from either direction of g of x, even though right at 2, the function is equal to 1, because it's discontinuous. The limit as we're approaching 2, we're getting closer, and closer, and closer to 4.