# 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.

i never struggled to understand anything in math. Except for this lesson. legit on my senior year my brain shut downs

your not telling limit correct defination

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

Thank you, this explanation helps me understand limits 🙂

Okay. This is epic.

is there any real world example where limit is used .. what is benefit of limit ?

that really helps me thx

sorry i can't learn about this because limits dont apply to me

mic drop bruv

and i thought 2 + x = 1 is bonkers.

btw i'm 15

what. are. you. talking about.

0 over 0 is not undefined. 0 / 0 = 0

0 is the only number that

canbe divided by 0 without resulting in an undefined value.~~at least that's what i've always been taught, i could totally be misinformed~~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

tnx sir

Who else is taking AP calc bc rn

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.

at 9:13 he farted i swear 😂

good job khan G

💚

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.

This guy should seriously be nominated for the Nobel Prize.

*year old kid

You need to add bangali caption

The limit doesnt exist

The concept could have explained much better than this.

Calculus makes me physically ill

I think 0/0 is indeterminate

Amazing 😍

This lecture is really helpful. Thank you so much

The digital calculator, how and where did you get that Mr. Khan?

Sir u r doing a great job

Math lover

Now I have a limited understanding about the limits

Me: I'm failing calculus

Sal: I'm gonna stop you right there

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

Türkiyeden gelmişem altyazı yazanın ellerine sağlık

Excellent

Nice

💖👏👏👏👏👏

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

Thank you! This is really helpful!

May God continue blessing you sir i understand alot of things

I'm almost true!

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.

Anybody else in school watching this

am i the only one who still doesn't understand

How to calculate limit x tends to 0 for (1+x)^(1/x)

Excellent

thanks aesop rock

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

Limits are the "I'm not touching you" of mathematics

anyone else in grade 5

Wait!!!! isn’t 0 divided by 0 indeterminate?

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

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

I would love to see the confused faces of people whom newton first told about calculus.hocam limit türev integral çok iyi çarpanlara ayırma da istiyorum

One point"nine nine nine nine nine……"😂😂

8:48

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

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?

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

That 4 from calculator makes it awkward lmao

0/0 is not undefined its undetermined.

lim

∆x=0.000001

THIS POWER! I FEEL IT!!! I FEEL KNOWLEDGIBLE!!!

This black background is bad

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

It looks something like this

Wonderful Sir

“One point NEIN NEIN NEIN NEIN”

1.9999999999999999^2 = 3.9999999999999996

DIDNT HELP AT ALL

What's the previous video i should watch?

Undefined or indeterminate

My guy sounds like Nick Fury

5:34 that's a "g" not an "f"

whaaaat

Time stamp 4:31

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

thank you.

Im not surprised this has 4,000,000 views

Nick fury is teaching limits to the avengers……

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

precalculus : " mind your limits "

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?

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

10:27 i love how ur calculator just rounded it to 4 hahaha

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

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

what does the G mean in G(x) D:

Isn’t 0/0 indeterminate not undefined

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.

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

1:13 really? Anything thing divided by zero is undefined??

The video is super awesome

2 plus 2 is 4 minus 1 das 3 quick mafs

Very very nice

Does anyone can explain to me how that 4 came?

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.

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?

Lim. Exam(you)= Pass

You->Sal

dear math

pls solve your own problems

thx