Calculus, all content (2017 edition)
Introduction to limits (old)
An older video of Sal introducing the notion of a limit. Created by Sal Khan.
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- Why do people say, "Find the LIMIT of a function as it approaches some number"? Why can't we just say, "Find the value of the function as it approaches this number"? What does it mean by taking the "limit" of a function?(11 votes)
- The limit may or may not be the same thing as the value of the function.
The limit is what it LOOKS LIKE the function ought to be at a particular point based on what the function is doing very close to that point. If the function makes some sudden change at that particular point or if the function is undefined at that point, then the limit will be different than the value of the function.(31 votes)
- I still don't understand after watching the video multiple times. Why is there a limit? What does it mean "as the limit of x approaches 0 or 2"? Why? Why? Why? I'm so confused.(6 votes)
- The concept is important because it explains to us about the function's behavior on an x-y plane. The concept of limits may not seem important at this time, but in jobs, especially physics of motion, deals a lot with limits. If you were to take a rocket, for example, and want to know where it is at a certain time (x), you can use limits to identify where it is at. For example, as the limit of x approaches 3 seconds, the rocket is approaches 50 kilometers.
A strong background in Algebra II and PreCalculus will solidify your knowledge of limits. Those subjects explain the basics of limits, and Calculus will show you some application of those limits (in continuity, rate of change, velocity.....and so much more).(7 votes)
- does the hole represent discontinuity in the function?(4 votes)
- If you are speaking of an empty circle on the graph, then, yes, that is how a discontinuity is depicted. Either the function is not defined at that point or else there is a sudden, discontinuous, jump in what the value of the function is at that point.(6 votes)
- Hello, can someone please give me some possible applications of limits in physics, engineering, etc. Just curious.(3 votes)
- Some specifc examples might also be right here in mathematics. If you remember from conic sections we actually use limits to find the assympthotes of the hyperbola, also you can use limits to define other cool stuff. For example, imagine that suddenly the formula to get the area of a circle has been removed from all knowledge, texbooks, etc... and you want to calculate the area of a circle. You might say "G, it's imposible!" but actually limits can help with this problem. If you take a polygon, let's say a square, you can put it inside the circle and the area of the square is going to be somehow close to the area of the circle, but still very far off from the real value. However as you add more and more sides to the polygon (imagine an hexagon inside of the circle) then the area of the polygon APPROCHES the area of the circle more and more. Now let's say "S" denotes the number of sides of the polygon, then you can define a function in wich to determine quite precisely the area of the polygon in terms of "S". So you could state that as "S" (The number of sides of the polygon) approches infinity, then the area of the polygon approches or is basically the same as the area of the circle. Pretty cool Uhh?
Hope this gives you an insight to what you can do with limits and really encourage you to keep learning about this topic in particular.
P.S. If you'd like a more solid explanation about the math involved to the problem of the circle and the polygon feel free to tell me and i'll work out the math.(5 votes)
- What is the purpose of limits? What do they help you find out about a function?(2 votes)
- let's say a function f(x) is undefined at a point, assume it to be 0. But you want to find the value very very very close to zero, in that case limits are useful.
Limits are also used to derive the derivative of a function(5 votes)
- hii..my question is that why the value of 0! and 1! is same.....till now i ask to many teacher but no one told me the reason....so please give me answer as soon as possible...because i very excite to know about it...(2 votes)
- Note that if n >= 1, then (n + 1)! / n! = [1*2*3*...n(n+1)] / (1*2*3*...*n) = n + 1.
We would like to extend this rule to n = 0, which would give 1!/0! = 1. Since 1! = 1, the definition 0! = 1 would follow.
Have a blessed, wonderful day!(3 votes)
- can a function have a circular graph?(0 votes)
- no it cant have a circular graph as for a single value of x there are 2 solutions which is not possible...(1 vote)
- Hi there.... if i multiply infinity with infinity, can i write it as infinity squared? And then can't we divide infinity by infinity?? Does both of them have any value?(1 vote)
- Infinity is not a proper number, so it has some special rules. Any positive number (except 0) multiplied by infinity is infinity, and that includes infinity, so
∞·∞ = ∞.
Also, any number divided by infinite tends to 0, so
a/∞ = 0
On the other hand, the division of infinity by infinity is undefined, as is the multiplication of infinity by 0:
∞/∞ = indeterminate,
∞·0 = indeterminate(4 votes)
- if you graph x^2 you get a parabola. what happens when you graph x^3? how about x^4? x^5? please help!(1 vote)
- Why don't you just plot them yourself and see what they look like? It's a good way to learn. Use a graphing calculator, or a spreadsheet program to make it faster. Even doing it by hand won't take very long.(2 votes)
- What does this mean?
- You will hear the term, "approaching x=2 from the left side." You can actually put numbers into the function to see how this works. Try solving the function using x=1.9 then 1.99 and 1.999. Repeat this pattern until you see what is happening. If the limit exists, you can find it by approaching from the (left) 2⁻ or the (right) 2⁺.(3 votes)
Welcome to the presentation on limits. Let's get started with some-- well, first an explanation before I do any problems. So let's say I had-- let me make sure I have the right color and my pen works. OK, let's say I had the limit, and I'll explain what a limit is in a second. But the way you write it is you say the limit-- oh, my color is on the wrong-- OK, let me use the pen and yellow. OK, the limit as x approaches 2 of x squared. Now, all this is saying is what value does the expression x squared approach as x approaches 2? Well, this is pretty easy. If we look at-- let me at least draw a graph. I'll stay in this yellow color. So let me draw. x squared looks something like-- let me use a different color. x square looks something like this, right? And when x is equal to 2, y, or the expression-- because we don't say what this is equal to. It's just the expression-- x squared is equal to 4, right? So a limit is saying, as x approaches 2, as x approaches 2 from both sides, from numbers left than 2 and from numbers right than 2, what does the expression approach? And you might, I think, already see where this is going and be wondering why we're even going to the trouble of learning this new concept because it seems pretty obvious, but as x-- as we get to x closer and closer to 2 from this direction, and as we get to x closer and closer to 2 to this direction, what does this expression equal? Well, it essentially equals 4, right? The expression is equal to 4. The way I think about it is as you move on the curve closer and closer to the expression's value, what does the expression equal? In this case, it equals 4. You're probably saying, Sal, this seems like a useless concept because I could have just stuck 2 in there, and I know that if this is-- say this is f of x, that if f of x is equal to x squared, that f of 2 is equal to 4, and that would have been a no-brainer. Well, let me maybe give you one wrinkle on that, and hopefully now you'll start to see what the use of a limit is. Let me to define-- let me say f of x is equal to x squared when, if x does not equal 2, and let's say it equals 3 when x equals 2. Interesting. So it's a slight variation on this expression right here. So this is our new f of x. So let me ask you a question. What is-- my pen still works-- what is the limit-- I used cursive this time-- what is the limit as x-- that's an x-- as x approaches 2 of f of x? That's an x. It says x approaches 2. It's just like that. I just-- I don't know. For some reason, my brain is working functionally. OK, so let me graph this now. So that's an equally neat-looking graph as the one I just drew. Let me draw. So now it's almost the same as this curve, except something interesting happens at x equals 2. So it's just like this. It's like an x squared curve like that. But at x equals 2 and f of x equals 4, we draw a little hole. We draw a hole because it's not defined at x equals 2. This is x equals 2. This is 2. This is 4. This is the f of x axis, of course. And when x is equal to 2-- let's say this is 3. When x is equal to 2, f of x is equal to 3. This is actually right below this. I should-- it doesn't look completely right below it, but I think you got to get the picture. See, this graph is x squared. It's exactly x squared until we get to x equals 2. At x equals 2, We have a grap-- No, not a grap. We have a gap in the graph, which maybe could be called a grap. We have a gap in the graph, and then we keep-- and then after x equals 2, we keep moving on. And that gap, and that gap is defined right here, what happens when x equals 2? Well, then f of x is equal to 3. So this graph kind of goes-- it's just like x squared, but instead of f of 2 being 4, f of 2 drops down to 3, but then we keep on going. So going back to the limit problem, what is the limit as x approaches 2? Now, well, let's think about the same thing. We're going to go-- this is how I visualize it. I go along the curve. Let me pick a different color. So as x approaches 2 from this side, from the left-hand side or from numbers less than 2, f of x is approaching values approaching 4, right? f of x is approaching 4 as x approaches 2, right? I think you see that. If you just follow along the curve, as you approach f of 2, you get closer and closer to 4. Similarly, as you go from the right-hand side-- make sure my thing's still working. As you go from the right-hand side, you go along the curve, and f of x is also slowly approaching 4. So, as you can see, as we go closer and closer and closer to x equals 2, f of whatever number that is approaches 4, right? So, in this case, the limit as x approaches 2 is also equal to 4. Well, this is interesting because, in this case, the limit as x approaches 2 of f of x does not equal f of 2. Now, normally, this would be on this line. In this case, the limit as you approach the expression is equal to evaluating the expression of that value. In this case, the limit isn't. I think now you're starting to see why the limit is a slightly different concept than just evaluating the function at that point because you have functions where, for whatever reason at a certain point, either the function might not be defined or the function kind of jumps up or down, but as you approach that point, you still approach a value different than the function at that point. Now, that's my introduction. I think this will give you intuition for what a limit is. In another presentation, I'll give you the more formal mathematical, you know, the delta-epsilon definition of a limit. And actually, in the very next module, I'm now going to do a bunch of problems involving the limit. I think as you do more and more problems, you'll get more and more of an intuition as to what a limit is. And then as we go into drill derivatives and integrals, you'll actually understand why people probably even invented limits to begin with. We'll see you in the next presentation.