Introduction to limits | Limits | Differential Calculus | Khan Academy

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.

Paul Whisler


  1. Thanks sir u are really making students lives simpler with this amazing teaching of yours .

  2. what. are. you. talking about.
    0 over 0 is not undefined. 0 / 0 = 0
    0 is the only number that can be divided by 0 without resulting in an undefined value.

    at least that's what i've always been taught, i could totally be misinformed

  3. i understood everything, thanks thanks

    personal notes for me…
    g(x) = y or y axis value

    in video,
    this is
    g(x) or y = { x^2 , x == 2; 1 , x = 2}
    this is a way of defining a function, here this one specifically means is

    if g of x or y is equal to x^2 then dont take 2 as the value of x, you can take any thing else but two.

    if g of x or y is equal to 1 at any point on function then value of x will be equal to 2

    thats just how function is defined

    date: wed/sept26/7:12am

  4. So here my internet connection has some serious limits.
    So what'll happen if I approach those limits?
    Let's say a limit of 5
    But as soon as I approach 5
    It'll be over
    Or no,even if I get close to 4
    It'll be still be over.

  5. For somebody who grew up and has worked in computer science for 30 years, this is confusing as hell. X, Y, Z? So the function of g(x) limit is… What the limit of that function for Y would be… At a certain constraint? I don't get it. Is that what is is? So X is always defined, but in Calculus, the function defines Y? Christ, this is so primitive. Not that I'm saying I'm so smart this is easy- quite the opposite. I feel like it's relearning concepts I should have been taught in the first place! Math is crazy, I will never learn 1% of it.

  6. Oh l am a ultra genius who dont need to understand this limits😂😂😂 cause I can't understand this limits 😂

  7. Thanks For Educating Worldwide! You Are My Favorite Personal Teacher! You Basically Have The Coolest Job!

  8. Dear Khan, I am pretty sure that you are a good teacher. I mean, I passed my test, you know… However, I can't stand your voice. You never change the tone of it while you are talking. Other than that, so proud of you fam. 4.5M subscribes. You must be making hella cash.

  9. Sat through 4 lectures on this, couldn’t grasp any of it, 5 minutes on YouTube and it’s already making sense lmao

  10. my mom used to watch these alot when i was in like 1st grade, im in 6th now and watching them myself lol

  11. Am I the only one whos annoyed that the pointer is writing and not a pen shaped? the thought that he is writing with a mouse (he might be not) annoys me

  12. I would love to see the confused faces of people whom newton first told about calculus.

  13. I know that you helped me and your video's are good and helpful to understand things easily

  14. ix+1/x+1 equal to limit of 0/0;
    and if u simplified it will equal to 1 why?
    the are the same expression, right?

  15. What just happened from 6:04 to 6:30…[PARABOLA]😂😂😂😂😂

  16. I lost it when he got 4. "Well it's not actually gonna be four" he says. Lol

  17. When im frustrated and about to give up and i hear Sals voice its instant reassurance and calmness. The way he approaches all topics is amazing

  18. this dude's actually really hilarious, 'not the most beautifully drawn parabola in the history of drawing parabolas'!!

  19. what's the name of that program that he's using to perform all of the drawing?

    I take it the calculator is part of it?

  20. is this what we learn in high school pre calc? somebody help plzzzzzz

  21. Thank you for this vid, when I get back to school I can explain it to my friends!

  22. let me pull out my trusty TI-85
    so whats 1.9999999 squared … oh its actually not going to be that, this calculator rounded things up

  23. I cannot begin to tell you how grateful I am for these videos. I'm a sophomore in an incredibly fast-paced calculus class and have gone home every day crying, desperately trying to solve my work with almost no understanding of it. This helps a lot, thank you.

  24. Careful, now…for 0/0 the function is undefined, but 0/0 is “indeterminate.”

  25. You guys are some pretty swaggy people. Whatever you're paid you still deserve a raise. Helping these people with math, you're basically heroes.

  26. Sir I’ve got a question.
    Should limits be simplified?
    Lets say f(x)= lim 2x+8-1/4x-2 wherein x=2

    My another question is.

    Is the special product operation applicable?

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