Let me draw a function

that would be interesting to take a limit of. And I’ll just draw it visually

for now, and we’ll do some specific examples

a little later. So that’s my y-axis,

and that’s my x-axis. And let;s say the function

looks something like– I’ll make it a fairly

straightforward function –let’s say it’s a line,

for the most part. Let’s say it looks just

like, accept it has a hole at some point. x is equal to a, so

it’s undefined there. Let me black that point

out so you can see that it’s not defined there. And that point there

is x is equal to a. This is the x-axis, this is

the y is equal f of x-axis. Let’s just say

that’s the y-axis. And let’s say that this

is f of x, or this is y is equal to f of x. Now we’ve done a bunch

of videos on limits. I think you have an

intuition on this. If I were to say what is the

limit as x approaches a, and let’s say that this

point right here is l. We know from our previous

videos that– well first of all I could write it down –the

limit as x approaches a of f of x. What this means intuitively is

as we approach a from either side, as we approach it from

that side, what does f of x approach? So when x is here,

f of x is here. When x is here, f

of x is there. And we see that it’s

approaching this l right there. And when we approach a from

that side– and we’ve done limits where you approach from

only the left or right side, but to actually have a limit it

has to approach the same thing from the positive direction and

the negative direction –but as you go from there, if you pick

this x, then this is f of x. f of x is right there. If x gets here then it goes

here, and as we get closer and closer to a, f of x approaches

this point l, or this value l. So we say that the limit

of f of x ax x approaches a is equal to l. I think we have that intuition. But this was not very, it’s

actually not rigorous at all in terms of being specific

in terms of what we mean is a limit. All I said so far is as

we get closer, what does f of x get closer to? So in this video I’ll attempt

to explain to you a definition of a limit that has a little

bit more, or actually a lot more, mathematical rigor than

just saying you know, as x gets closer to this value, what

does f of x get closer to? And the way I think about it’s:

kind of like a little game. The definition is, this

statement right here means that I can always give you a range

about this point– and when I talk about range I’m not

talking about it in the whole domain range aspect, I’m just

talking about a range like you know, I can give you a distance

from a as long as I’m no further than that, I can

guarantee you that f of x is go it not going to be any further

than a given distance from l –and the way I think about it

is, it could be viewed as a little game. Let’s say you say, OK Sal,

I don’t believe you. I want to see you know, whether

f of x can get within 0.5 of l. So let’s say you give me 0.5

and you say Sal, by this definition you should always

be able to give me a range around a that will get f of

x within 0.5 of l, right? So the values of f of x are

always going to be right in this range, right there. And as long as I’m in that

range around a, as long as I’m the range around you give me, f

of x will always be at least that close to our limit point. Let me draw it a little bit

bigger, just because I think I’m just overriding the same

diagram over and over again. So let’s say that this is f of

x, this is the hole point. There doesn’t have to be a hole

there; the limit could equal actually a value of the

function, but the limit is more interesting when the function

isn’t defined there but the limit is. So this point right here– that

is, let me draw the axes again. So that’s x-axis, y-axis x,

y, this is the limit point l, this is the point a. So the definition of the limit,

and I’ll go back to this in second because now that it’s

bigger I want explain it again. It says this means– and this

is the epsilon delta definition of limits, and we’ll touch on

epsilon and delta in a second, is I can guarantee you that

f of x, you give me any distance from l you want. And actually let’s

call that epsilon. And let’s just hit on

the definition right from the get go. So you say I want to be no more

than epsilon away from l. And epsilon can just be any

number greater, any real number, greater than 0. So that would be, this distance

right here is epsilon. This distance there is epsilon. And for any epsilon you give

me, any real number– so this is, this would be l plus

epsilon right here, this would be l minus epsilon right here

–the epsilon delta definition of this says that no matter

what epsilon one you give me, I can always specify a

distance around a. And I’ll call that delta. I can always specify

a distance around a. So let’s say this is delta

less than a, and this is delta more than a. This is the letter delta. Where as long as you pick an x

that’s within a plus delta and a minus delta, as long as the x

is within here, I can guarantee you that the f of x, the

corresponding f of x is going to be within your range. And if you think about it

this makes sense right? It’s essentially saying, I can

get you as close as you want to this limit point just by– and

when I say as close as you want, you define what you want

by giving me an epsilon; on it’s a little bit of a game

–and I can get you as close as you want to that limit point by

giving you a range around the point that x is approaching. And as long as you pick an x

value that’s within this range around a, long as you pick an x

value around there, I can guarantee you that f of x will

be within the range you specify. Just make this a little bit

more concrete, let’s say you say, I want f of x to be within

0.5– let’s just you know, make everything concrete numbers. Let’s say this is the number 2

and let’s say this is number 1. So we’re saying that the limit

as x approaches 1 of f of x– I haven’t defined f of x, but it

looks like a line with the hole right there, is equal to 2. This means that you can

give me any number. Let’s say you want to try it

out for a couple of examples. Let’s say you say I want f of x

to be within point– let me do a different color –I want f

of x to be within 0.5 of 2. I want f of x to be

between 2.5 and 1.5. Then I could say, OK, as long

as you pick an x within– I don’t know, it could be

arbitrarily close but as long as you pick an x that’s –let’s

say it works for this function that’s between, I don’t

know, 0.9 and 1.1. So in this case the delta from

our limit point is only 0.1. As long as you pick an x that’s

within 0.1 of this point, or 1, I can guarantee you that your

f of x is going to lie in that range. So hopefully you get a little

bit of a sense of that. Let me define that with the

actual epsilon delta, and this is what you’ll actually see in

your mat textbook, and then we’ll do a couple of examples. And just to be clear, that

was just a specific example. You gave me one epsilon and I

gave you a delta that worked. But by definition if this is

true, or if someone writes this, they’re saying it doesn’t

just work for one specific instance, it works for

any number you give me. You can say I want to be within

one millionth of, you know, or ten to the negative hundredth

power of 2, you know, super close to 2, and I can always

give you a range around this point where as long as you pick

an x in that range, f of x will always be within this range

that you specify, within that were you know, one trillionth

of a unit away from the limit point. And of course, the one thing

I can’t guarantee is what happens when x is equal to a. I’m just saying as long as you

pick an x that’s within my range but not on a, it’ll work. Your f of x will show up to be

within the range you specify. And just to make the math

clear– because I’ve been speaking only in words so far

–and this is what we see the textbook: it says look, you

give me any epsilon greater than 0. Anyway, this is a

definition, right? If someone writes this they

mean that you can give them any epsilon greater than 0, and

then they’ll give you a delta– remember your epsilon is how

close you want f of x to be to your limit point, right? It’s a range around f of x

–they’ll give you a delta which is a range

around a, right? Let me write this. So limit as approaches a

of f of x is equal to l. So they’ll give you a delta

where as long as x is no more than delta– So the distance

between x and a, so if we pick an x here– let me do another

color –if we pick an x here, the distance between that value

and a, as long as one, that’s greater than 0 so that x

doesn’t show up on top of a, because its function might be

undefined at that point. But as long as the distance

between x and a is greater than 0 and less than this x

range that they gave you, it’s less than delta. So as long as you take an x,

you know if I were to zoom the x-axis right here– this is a

and so this distance right here would be delta, and this

distance right here would be delta –as long as you pick an

x value that falls here– so as long as you pick that x value

or this x value or this x value –as long as you pick one of

those x values, I can guarantee you that the distance between

your function and the limit point, so the distance between

you know, when you take one of these x values and you evaluate

f of x at that point, that the distance between that f of x

and the limit point is going to be less than the

number you gave them. And if you think of, it seems

very complicated, and I have mixed feelings about where

this is included in most calculus curriculums. It’s included in like the, you

know, the third week before you even learn derivatives, and

it’s kind of this very mathy and rigorous thing to think

about, and you know, it tends to derail a lot of students and

a lot of people I don’t think get a lot of the intuition

behind it, but it is mathematically rigorous. And I think it is very valuable

once you study you know, more advanced calculus or

become a math major. But with that said, this

does make a lot of sense intuitively, right? Because before we were talking

about, look you know, I can get you as close as x approaches

this value f of x is going to approach this value. And the way we mathematically

define it is, you say Sal, I want to be super close. I want the distance to be

f of x [UNINTELLIGIBLE]. And I want it to be

0.000000001, then I can always give you a distance around x

where this will be true. And I’m all out of

time in this video. In the next video I’ll do some

examples where I prove the limits, where I prove some

limit statements using this definition. And hopefully you know, when we

use some tangible numbers, this definition will make a

little bit more sense. See you in the next video.

The potato quality hurts

SUPPPERRRR CLOSE

Screw school. Khan Academy is my new way.

youre going to heaven

I understand that we start with an epsilon…but why do we start with an epsilon? That never quite made sense to me…

I understood the idea behind, but I couldn't understand how sal derived that linear inequality equation…

To understand that equation I have even gone through the linear inequality playlist but still its vague for me😞😞😞😞😞

Thnx to khan academy …Finally I got it..😍

Great viedeo

Thanks sir.

This won't be valid if function is not defined in Neighborhood of A right?

worst explntn

First day at college, and I have to deal with this.

Helped a lot for my college. This thing hardly went through my brain while in class, lmao. Especially its problems lol

Don't you need to specify that if the delta interval gets smaller, the epsilon interval must get smaller???

What if delta can get really small but epsilon has to stay big. Then this definition say that it's still a limit, when the function might not be approaching anything.

Best explanation ever 😃😃

This was very hard to understand

need to update this video

i don"t understand that video..plz make another good video about it. thanks #Sal

thanks for the video, learning this on my own, i think you helped me somewhat acquire the understanding of this subject.

12:22

Sal Khan – One of best people on the internets, period. He explains it far, far better than my $120,000 professor ever could.

I thought I understood math until I started studying Analysis

Get to the point already lad

If you pick a range that limx->a f(x) falls into on the y axis greater than 0, there is a corresponding positive range about a on the x axis, if the limit exists.

If the limit does not exist, you will never find a corresponding range about a on the x axis no matter how small you pick the range about f(x) on the y axis.

Is this where convergent/divergent limits come from? As Epsilon gets smaller, delta gets smaller. If delta gets bigger as e gets smaller, the limit diverges: the limit does not exist.

you kinda sound like jimi hendrix

My math book stated that the distance between f(x) and L are less than the epsilon divided by 2? Can anyone help me on that plsss

10 years later this video still better than the lecture I got in class…

thx!

second week in calc I… 8 wk course.

thank god for resources like this. i remember when people that were good at math kept all the information secret.

when you're taking calculus over the summer, and he says they teach you this over the first three weeks and my professor gave it to me the first day 🙂

Thank god you exist

Thanks!

I love u

We learned it on the second day of BC 🤧🤧🤧🤧

At 4:55 Sal is pretty much like, “We’ll talk about epsilon and delta later, nvm we’re taking about them now” 😂😂

12:23 when he hittin it from the back and you haven’t done anal in a long time

Here’s one thing I don’t understand: why is are the distances before and after the limit L equal (epsilon), while the distances before and after the c value (delta) are also equal? This seems to fit more accurately a line than a curve where the rate of change changes. If you pick 2 f(x) values on a curve of equal distance from L and then find their corresponding x values, those x values will not be equidistant from the c value that corresponds to the limit L.

I’m having trouble getting past this.

Thank you so much. I might just pass my quiz thanks to you.

If we pick any value which is less than the delta (means within the range of the delta value) then the distance of f(x) and the limit is less than the epsilon. Does this means that if we substitute the delta value, it will have an f(x)=Epsilon?

Wowwww, explained very simply

Viewing after 10 yrs of upload when sal is not making vids🥞

I'm doing dentistry but because it's not my passion and I'm only doing it for fear of staying poor in the future I'm learning physics and math at home, you guys help me more than anything else