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Limits intro

Limits describe how a function behaves near a point, instead of at that point. This simple yet powerful idea is the basis of all of calculus.
To understand what limits are, let's look at an example. We start with the function f(x)=x+2.
The limit of f at x=3 is the value f approaches as we get closer and closer to x=3. Graphically, this is the y-value we approach when we look at the graph of f and get closer and closer to the point on the graph where x=3.
For example, if we start at the point (1,3) and move on the graph until we get really close to x=3, then our y-value (i.e. the function's value) gets really close to 5.
Created with Geogebra.
Similarly, if we start at (5,7) and move to the left until we get really close to x=3, the y-value again will be really close to 5.
Created with Geogebra.
For these reasons we say that the limit of f at x=3 is 5.
You might be asking yourselves what's the difference between the limit of f at x=3 and the value of f at x=3, i.e. f(3).
So yes, the limit of f(x)=x+2 at x=3 is equal to f(3), but this isn't always the case. To understand this, let's look at function g. This function is the same as f in every way except that it's undefined at x=3.
Just like f, the limit of g at x=3 is 5. That's because we can still get very very close to x=3 and the function's values will get very very close to 5.
So the limit of g at x=3 is equal to 5, but the value of g at x=3 is undefined! They are not the same!
That's the beauty of limits: they don't depend on the actual value of the function at the limit. They describe how the function behaves when it gets close to the limit.
Problem 1
This is the graph of h.
What is a reasonable estimate for the limit of h at x=3?
Choose 1 answer:

We also have a special notation to talk about limits. This is how we would write the limit of f as x approaches 3:
"The limit of""the function f"limx3f(x)"as x approaches 3."
The symbol lim means we're taking a limit of something.
The expression to the right of lim is the expression we're taking the limit of. In our case, that's the function f.
The expression x3 that comes below lim means that we take the limit of f as values of x approach 3.
Problem 2
This is the graph of f.
What is a reasonable estimate for limx6f(x) ?
Choose 1 answer:

Problem 3
Which expression represents the limit of x2 as x approaches 5?
Choose 1 answer:

In limits, we want to get infinitely close.

What do we mean when we say "infinitely close"? Let's take a look at the values of f(x)=x+2 as the x-values get very close to 3. (Remember: since we're dealing with limits we don't care about f(3) itself.)
xf(x)
2.94.9
2.994.99
2.999close to 34.999close to 5
We can see how, when the x-values are smaller than 3 but become closer and closer to it, the values of f become closer and closer to 5.
xf(x)
3.15.1
3.015.01
3.001close to 35.001close to 5
We can also see how, when the x-values are larger than 3 but become closer and closer to it, the values of f become closer and closer to 5.
Notice that the closest we got to 5 was with f(2.999)=4.999 and f(3.001)=5.001, which are 0.001 units away from 5.
We can get closer than that if we want. For example, suppose we wanted to be 0.00001 units from 5, then we would pick x=3.00001 and then f(3.00001)=5.00001.
This is endless. We can always get closer to 5. But that's exactly what "infinitely close" is all about! Since being "infinitely close" isn't possible in reality, what we mean by limx3f(x)=5 is that no matter how close we want to get to 5, there's an x-value very close to 3 that will get us there.
If you find this hard to grasp, maybe this will help: how do we know there are infinite different integers? It's not like we've counted them all and got to infinity. We know they are infinite because for any integer there's another integer that's even larger than that. There's always another one, and another one.
In limits, we don't want to get infinitely big, but infinitely close. When we say limx3f(x)=5, we mean we can always get closer and closer to 5.
Problem 4
xg(x)
7.16.32
7.016.1
7.0016.03
6.9996.03
6.996.1
6.96.32
What is a reasonable estimate for limx7g(x)?
Choose 1 answer:

Another example: limx2x2

Let's analyze limx2x2, which is the limit of the expression x2 when x approaches 2.
We can see how, when we approach the point where x=2 on the graph, the y-values are getting closer and closer to 4.
Created with Geogebra.
We can also look at a table of values:
xx2
1.93.61
1.993.9601
1.999close to 23.996001close to 4
xx2
2.14.41
2.014.0401
2.001close to 24.004001close to 4
We can also see how we can get as close as we want to 4. Suppose we want to be less than 0.001 units from 4. Which x-value close to x=2 can we choose?
Let's try x=2.001:
2.0012=4.004001
That's more than 0.001 units away from 4. Alright, so let's try x=2.0001:
2.00012=4.00040001
That's close enough! By trying x-values that are closer and closer to x=2, we can get even closer to 4.
In conclusion, limx2x2=4.

A limit must be the same from both sides.

Coming back to f(x)=x+2 and limx3f(x), we can see how 5 is approached whether the x-values increase towards 3 (this is called "approaching from the left") or whether they decrease towards 3 (this is called "approaching from the right").
Now take, for example, function h. The y-value we approach as the x-values approach x=3 depends on whether we do this from the left or from the right.
When we approach x=3 from the left, the function approaches 4. When we approach x=3 from the right, the function approaches 6.
When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.
Problem 5
This is the graph of function g.
Which of the limits exists?
Choose all answers that apply:

Want to join the conversation?

  • blobby green style avatar for user boddhulas
    In the last question, how does
    x→7
    ​lim
    ​​ g(x)
    exist? It has two locations right?
    (135 votes)
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    • duskpin ultimate style avatar for user Rachel
      The limit exists because the same y-value is approached from both sides. It does not have two locations because the open circle is a just gap in the graph. The closed circle is the actual y-value for when x=7.
      (302 votes)
  • blobby green style avatar for user K M
    In problem 5 why can one of the answers be x→6? but not x→3?
    (19 votes)
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    • blobby green style avatar for user Dave
      I assume you are talking about the last example. As you approach x=6 from the left you move closer to 3; AND as you approach x=6 from the right, you also moce closer to 3.

      As you move closer to x=3 form the left you move closer to 3, BUT when you move closer to x=3 from the right, you move closer to 6.

      They must be moving to the same value of y from both sides if you are not going to specify the side in th limit notation.
      (36 votes)
  • primosaur seedling style avatar for user Amanda Lu
    For the last question, how is there a limit for x-->6? There is a point there. I thought there could only be limits if there were open dots.
    (11 votes)
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    • male robot hal style avatar for user Stephen.Burke
      As Sal explained both in the video Limits intro, and in the text, the beauty of limits, and one property of limits, is that they do not explain the actual point of the graph, but the behavior leading up to that point. Whether there is a point at f(x)= 6 or a hole, that would not change that there still is a limit, unless a jump occurs between the two leading lines.
      (43 votes)
  • eggleston blue style avatar for user sam
    learning this calculus concept here was actually super fun. although i know this will take many hours, the hours will be exhilarating! i'm going to try and conquer calculus in 2 weeks. (today is june 24th, i have until july 10th)
    (22 votes)
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  • blobby green style avatar for user 1249424990bsq
    what is the difference between APcalculus AB and APcalculus BC?
    (7 votes)
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    • leaf grey style avatar for user Alex
      More specific course content is given on the College Board website.

      Essentially, AB is equivalent to Calc 1, while BC is equivalent to Calc 1 and 2. AB covers limits, derivatives, and integrals. BC covers everything that AB does, in addition to derivatives of vector-valued functions, polar functions, parametric functions, planar motion, Euler's Method, improper integrals, integration by parts, arc length, polar areas, the logistic model, and (a whole unit on) series. Hope that I helped.
      (22 votes)
  • blobby green style avatar for user cr7neymar9
    over a year ago is was on algebra 1 videos now I'm on calculus finally. lets goo
    (17 votes)
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    • blobby green style avatar for user mezomaxim516
      Bro what? Congratulations, but be sure you have all the fundamentals down from Algebra 1 and 2 along with Geometry and precal. You don't want to jump ahead into something that you don't understand. Be sure to have strong foundations before continuing.
      (0 votes)
  • blobby green style avatar for user mzhang
    the plot has open dots and closed dots. what does that symbolize in limits?
    (5 votes)
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  • leaf red style avatar for user Aiden Houser
    On problem 5, there are limits even though the function is not undefined right? For example, as x approaches 6
    (5 votes)
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    • male robot donald style avatar for user Venkata
      Correct. If a function is defined at a point, there can be two cases:

      1. It is defined but not continuous (like at x = 7)

      2. It is defined and continuous (like at x = 6)

      Note that even though the function is continuous and the limit exists at x = 6, it is not differentiable, a term you'll learn later on. Revisit this answer once you learn differentiability. You'll see what I mean
      (12 votes)
  • leaf orange style avatar for user curiousfermions
    When we take a limit, we approach a specific x value from both sides. But there is an infinite number of numbers between any two numbers. So when we are approaching x, can we ever really get there? And if we can not, then does the calculus give us an answer with really really really really really small flaw that does not create problem in any calculation? Or is it 100% flawless?
    (7 votes)
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    • mr pants teal style avatar for user Moon Bears
      This is the issue that a lot of people had in the development of the calculus. The solution was quite clever, the idea is that you can get "arbitrarily close" that is given any epsilon positive there's a delta such that |x-a| < delta implies |f(x) - f(a)| < epsilon. If this is true FOR EVERY epsilon, then we say the limit exists and the function is continuous. Is it 100% flawless? Not really, but the thing is it's useful. Using these ideas we were able to build up a foundation for calculus which eventually lead to the physics that put man on the moon. The point is it gets results.
      (5 votes)
  • primosaur seedling style avatar for user Manaf01
    first day learning calculus
    (10 votes)
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