Introduction to limits | Limits | Differential Calculus | Khan Academy

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Courses on Khan Academy are always 100% free. Start practicing-and saving your progress-now: www.khanacademy.org/math/ap-c...
Introduction to limits
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Differential calculus on Khan Academy: Limit introduction, squeeze theorem, and epsilon-delta definition of limits.
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Not all heroes wear capes

[@agustdel5571]

oh so true

[@Rhyff]

How do you know that he's not wearing a cape? ;)

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Not all heroes wear underwear over their pants....

[@JoeCnNd]

cool man!

[@gratefuldeadmau5610]

MalAware all of the cool ones do

[@Jack-gu4fc]

I've already learned this like 8 years ago, my job doesn't even require math. I just came here to watch a legend

[@macdavidsolomon9098]

So why are you telling us

[@rayoski241]

Mac David Solomon he’s telling you how much he admires his lecture. I am in the same boat. people that post stuff like this deserve praise. This video series help me a lot.

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No one cares. Paisa kama jake

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@@GantavyaDhandaB I care bro

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@@xander7001 me too i care bro :)

[@RajSankla123]

Me: I'm failing calculus Sal: I'm gonna stop you right there

[@BharCode09]

Lim. Exam(calculus)= Passs You->Sal

[@aryyancarman705]

@@BharCode09 lol

[@willr455]

Your parabolas are beautiful, don't let ANYBODY tell you otherwise.

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willr455 Real parabolas have curves.

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Real people don't judge a parabola by its curves, but by its personality :)

[@SathvickSatish]

If anyone didn't get Red Wolf's comment, search up "There's only one true parabola" and click on Matt's video

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@Teringventje Stfu

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Benson luo when graphing quadratic functions you get a parabola this U shape

[@statixc]

I haven't done advanced math since I left college. This is incredibly clear. My professors never made it this clear. Keep it up.

[@electric_deer2007]

Wait this is advanced math ??

[@WalterBlacc]

@@electric_deer2007 it's a beginning

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when you don't even understand khan academy, you know all hope it's lost 😭😭😭😭

[@relativemotion]

+Domonique Thompson Maybe you should revisit some of the lower level content first? Idk what to tell ya. I've gotten farther in mathematics via Khan Academy than I ever did in K-12 schooling (USA).

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at least im not the only one

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Same here 😭

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😭

[@shannenlibres2365]

@@relativemotion can you explain to me why in 7:08 the gap/limit is on positive y4? how did he know it had to be on point (x2, y4)? I understand the x2 part but not the y4 part. Also why is it that we define the entire parabola as "x^2" (from x^2, x≠2) not equal to when 1 (from 1,x=2) is just the placement on the y axis? I hope I make sense please please help me

[@ahmde]

Mind officially blown from the quality of this explanation.

[@shannenlibres2365]

sorry to bother you can you explain to me why in @ the gap/limit is on positive y4? how did he know it had to be on point (x2, y4)? I understand the x2 part but not the y4 part. Also why is it that we define the entire parabola as "x^2" (from x^2, x≠2) not equal to when 1 (from 1,x=2) is just the placement on the y axis? I hope I make sense please please help me

[@tttsss12350]

@@shannenlibres2365 well at first you just draw the graph of g(x) = x^2. Now it's given that x is not defined at 2 so you just omit that coordinate where x=2. It's just like you're painting a white wall of your house into any other color, than you realize that ohh i didn't want to color this part of the wall and you just paint it white again.

[@jameserayburn]

I really enjoy a good refresher. My grandmother was a high school teacher for 43 years, so I know that it takes a great person to be a great teacher, and Mr. Khan is one of those few great teachers.

[@shannenlibres2365]

sorry to bother you can you explain to me why in 7:08 the gap/limit is on positive y4? how did he know it had to be on point (x2, y4)? I understand the x2 part but not the y4 part. Also why is it that we define the entire parabola as "x^2" (from x^2, x≠2) not equal to when 1 (from 1,x=2) is just the placement on the y axis? I hope I make sense please please help me

[@afonsodeportugal]

@@shannenlibres2365 It's been a year. Did you figure it out?

[@nihilsinedeonihilsinedeo3137]

@@shannenlibres2365 because that's just how the function is defined. When x=2, in other words , f(2)= 1 not 4 because the value of the function when x=2 is 1. The x axis represents the values of x, the y axis represents the values of the function. Because of what I previously said , 4 can not be a value of x=2 , because f(2)=1 but can be a value of f(-2) because -2 is not equal to 2 , therefore f(-2)=4.

[@AcademiaVasquez]

Khan, can you please let me know what software you are using? I'm pretty sure it's not SmoothDraw anymore (since I use that myself). I am redoing your vids in Spanish. Thanks!

[@purab9]

look where u at now, 207K subs, bruh.

[@AnuragKumar-xh3wc]

Wow, I actually checked your channel and it is great. Why is not even verified?

[@Cosplayinghuman]

@@purab9 LoL I thought why he's on top in comments

[@purab9]

@@Cosplayinghuman lol your pfp

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@@AnuragKumar-xh3wc you have to request

[@DrTNTyt]

It is hard to learn about limits because my internet connection is limited

[@emperorpalpatine9841]

Hahaha, then it should be easy.

[@olivialew8175]

Its hard to learn about limits because my brain cells are limited

[@xxx_jim_the_reaper_xxx]

@@olivialew8175 Even with limited brain cells, you can play RPGs that also use numbers as well.

[@humanaccount2754]

What about sit in rooms where the connections is approaching the limit?

[@darkscienceyt]

I'm so happy that I finished college and dont have to watch these videos anymore...

[@commonemail4153]

Congrats!! ( I know its an old comment, you may be married now)

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eric 123 fr

[@crux7534]

But I challange that u will see these types of video again if u get any doubt

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😭 lucky u. Congrats

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@@commonemail4153 congrats bro🔥🔥

[@wolfieswifty13]

You are a hero! I have a quiz tomorrow on functions, limits, and derivatives and I knew nothing! At least now I know the basics. Thank you so much.

[@LearnMathsFree]

+Wolfie Swifty How did the quiz go?

[@brandonstreets9177]

did you do well?

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+Wolfie Swifty Did you pass your quiz?

[@K40TIKsPl4Yer]

+Wolfie Swifty We will NEVER know what heppened during this quizz

[@e02_v41]

+Wolfie Swifty he prolly got a 60 or something, one video isnt gonna do shit to prepare for a quiz he doesn't know anything bout

[@verabh]

I find it so helpful how these videos don't have fluff in the beginning and end, they just get straight to the point and don't waste any of your time.

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yes

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Nobody: Sal: Let me write it in a different color.

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m e m e s

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im a junior this year and learned basically nothing my sophmore year so im spending my summer watching your channel and actually oddly enjoying actually understanding math and pre calculus

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same

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you do calculus in junior year?????

[@newq]

I never enjoyed math until Sal Khan came along.

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Yeah, we do calculus in our junior in year( In India at the very least)

[@padmeamidala278]

saraa I did calculus junior year because I got ahead in math freshman year

[@williamj.stockich]

Calculus students don’t fret if you don’t grasp limits. They are only proofs that calculus is valid. Unless you are a mathematics major, you will only encounter limits superficially in other scientific courses. What you are being taught now was not developed until the 1880s. This means that all the progress with calculus from the time of Newton and Leibniz in the 1680s was achieved without any satisfactory validating proofs as to why calculus worked. Derivatives can be achieved according to a set of rules; integrals have fewer useful rules; limits have no rules. Each limit problem requires a different approach, and examples provided are usually too simple to help. When you reach the point of not knowing how to proceed, you are in a pool of millions. Don't get discouraged, the rest of calculus is much easier to understand.

[@binzsta86]

I'm not discouraged now, but will be when I bombed my exam.

[@rachelpilato6474]

THANK YOU! Starting calc in about a month and haven't done precalc since 2006 and I'm FREAKING THE HELL OUT and trying to determine what to focus on for review.

[@eldersprig]

There are 3 or 4 texts on infinitesimal calculus which go directly to the definitions of the D, by augmenting the Reals with infinitesimals. They then put 150 pages on limits into the appendix, but that's neither here nor there. Fluxions (infinitesimals) are intuitive but had no rigor, and it took a few centuries for someone to put the rigor into them, but then the damage was done so we're stuck with limits.

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Taking AP Calc AB this year. I'm absolutely terrified... but Khan Academy is truly a blessing and makes me feel more confident.

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fr dude... ap calc ab is saved because of khan academy :)

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Khan your a lucky guy. There are probably girls that are in love with you just because of your awesome teaching style and your voice.

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Mr. Khan does not make videos.. These videos are made by any one out there. Even we can make a video and send it.

[@IxMeTutorials]

Reva DS What? They are made by him. All of those videos have the same voice and it's Mr Khan's voice.

[@jonny5825]

IxMeTutorials No, Reva DS is right, these ones all just happen to be by the same guy, but at Khan Academy there are many, many, different instructors from all over the world.

[@IxMeTutorials]

Jonny Bloom Thats true but this particular video is actually voiced by Mr Khan himself.

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and you probably saved them from failing multiple times lmao

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Thanks. I'm a fetus, and this helps a lot!

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welcome to the world, ambitious little one

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how does the womb feel there lol

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I'm a zygote and I understood everything!

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some people are just born with calculus talent!

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@@malamalini9784 i havent even reached the egg

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This teacher is definitely the god sent soul to help a million of students in maths. I wish I should have seen this channel earlier….

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we need to understand that we are equally as capable of understanding something as our teachers but for knowledge to be transferred whole from one mind to the other, perfect communication is the key

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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

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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.

[@richchickenpanini7597]

I just watched a grown man struggle to draw a parabola

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isaac newton was a goddamn genius

[@The112Windows]

Gets bored. Invents Calculus. Issac Newton in a nutshell.

[@theflaggeddragon9472]

+Andrew Deen Hi didn't actually deal with or prove limits XD. But a genius indeed

[@nischithkumar2502]

Andrew Deen for some reason, your comment made me laugh

[@thehardyboys4227]

Yes between quarantining himself from the bubonic plague for 10 years and living that great life of celibacy he was able to master calculus.

[@jcers]

Lol in the words of Neil degrasse Tyson: “he invented calculus. Then he turned 26.”

[@cloudhair5921]

I have a maths exam in 4 days and this really helped me understand limits, part of what my teacher calls: 'conceptual understanding' and expects us to know. Thank you so so so much.

[@scrubnub4life]

yo, i got a final in 3 hours... THIS IS PERFECT

[@barackputin]

how did it go

[@standowner6979]

@@barackputin They failed!

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@@barackputin bruh it was 8 years ago😭😭

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Comments 7 years ago: “omg thank you for saving me i have quiz tomorrow” Comments now: “oh look dapz just ratioed khan academy”

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can’t believe this got ratio’d by dapz

[@mikuplaysleague8704]

I love this channel... It's so easy to understand. I learn so much from this channel, much more than my college professor. Thank you for pushing out such great content!

[@Reivivus]

Yeah, you explained what my Calculus teacher confounded me with for 3 hours. I didn't grasp anything in that class today...but, you are a true teacher...not an expecter!

[@columbus8myhw]

I remember, when I first learned this concept, I just thought of it as asking, "Where is the hole?"

[@columbus8myhw]

Lars Magnus Samuelsson Svenssonsenn Magnussvensamuelsson Thor Nah, I've had lots of practice.

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columbus8myhw ....with your mom

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Everytime I have to search up khan academy yk it’s gonna be a bad semester.

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10:34 Calculator:HAHA! GET REKT M8

[@ahmedyafeai6592]

life saver!!! I started calculus and the teacher skipped through the limits and i over thought it I knew what the limit or "the closer to" meant, but didn't know how to implement it into the graph.

[@nafsinbelay957]

Greatest educator of all time.

[@mr.coffee5220]

My favorite thing about Sal is he gives me asmr feelings, but he teaches me.

[@kalina.manova]

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.

[@janak7840]

Kalina Manova ur a sophomore in a calc class?? woww are u like advanced or something? i’m a junior in ap calc and i’m already struggling

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I am in for quite an adventure.

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You are a life saver. Whenever I need help for Calculus, I will always go to you since you can help me get out of the dumps.

[@TheSuperkiller14]

This guy is a legend. You really do deserve an award :)!

[@jessegabriel1438]

Calculus is infact so easy - only thing is: thinking its difficult before even studying it

[@shannenlibres2365]

sorry to bother you. can you explain to me why in 7:08 the gap/limit is on positive y4? how did he know it had to be on point (x2, y4)? I understand the x2 part but not the y4 part. Also why is it that we define the entire parabola as "x^2" (from x^2, x≠2) not equal to when 1 (from 1,x=2) is just the placement on the y axis? I hope I make sense please please help me. I'm swallowing my pride and asked so many people in the comments here in case nobody responds to me. Nobody has yet.

[@chagrin6919]

@@shannenlibres2365 i know I'm 2 weeks late but, the gap signifies that that particular value of x does not satisfy the equation of y. So we know that at x=2, y does not equal to x^2, instead y=1 at x=2. Thus gap is inserted in the parabola at y=4 to show that that particular equation (y=x^2) does not satisfy at x=2. Hope it helps.

[@sxehitman007]

I just can't tell you how thankful I am, for people like you Sal.

[@anjalinetragaonkar3738]

Did anyone else notice that, Sal's got an amazing voice or is it just me?

[@Anish-Kumar-Verma]

Han jab kisi ke ya kisi ki awaj se familiar ho jane pe accha hi lgta h 😁😂

[@alexpena9927]

this stuff is a life saver when calculating time complexity in my computer science courses

[@GjerdanPeterson]

the first example would've been easier to understand if both of the variable x and y were not 1 cause that confuses people...

[@topnaman2735]

I agree to that

[@adenc.9048]

Or if it was x-2/x-2

[@shannyoh03]

honestly this kind of videos is a lot better than my teacher.

[@destrisusilaningrum180]

this really helps me in providing a visual image to make my students understand about limit and continuity, thank you very much

[@echoesofsilence22]

Really, I'm in the 9th grade and finished watching this video with a basic knowledge of calculus. Thanks Sal, you're a hero.

[@maxgeorge1463]

ummmmm... yeah this isnt an intro to all of calculus buddy. This is a good fundamental of calc, but only a raindrop in the ocean of calculus. Watch 3Brow1Blue's vid compilation on calculus. That vid series pretty much introduces every large concept youll learn in Calc 1. Limits barely introduce applications of derivatives and integrals.

[@ricardomilos1784]

have you completed all calc classes?

[@maxgeorge1463]

@@ricardomilos1784 ??? All Calc classes? Specifications please

[@raicyceprine8953]

That joke about parabola made things lighter for me😂. It just makes me feel at ease in this serious topic

[@otiasahlivingstone1648]

My lectures is making it looks difficult and class so bored... Just find out from a friend and after watching few videos I think have leant a lot... Kudos Khan Academy...

[@TheGreatBrendon]

Sal, you are the greatest teacher there is, thank you for all your time.

[@AG-po7bl]

Brilliant explanation! You are a really talented teacher.

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Bruh, when Dapz Academy is the new pro 🤣🤣😂🤣😂🤣😂

[@CristianDMusic]

man, khan academy guy sold his soul, can't imagine how much effort all these lessons take. i know it's over the course of a long time but still, doing a great service nevertheless

[@Marshall...-_-1101]

Hats off to Khan Academy

[@mukhzarramaalik6349]

This lecture is really helpful. Thank you so much

[@shivangi3030]

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

[@mesqueunclub101]

SAL! Seriously i dont know how i would have made the exams without you! I am sure to pass Chemistry and calculus becouse of u. I am sure to pass biology, if i watch all of ur videos about biology! I owe u big time! Tnx man!

[@Phatency]

I love how Khan got back into making calculus videos

[@slothguy5716]

Watching these is the best way to review for an exam.

[@leosparton622]

I got exams in 20 minutes and i am hypercramming this helped i will tell you how it went when i am done

[@Ansh._.]

How it went??

[@leosparton622]

@@Ansh._. I failed horrifically however I passed the limits section

[@Ansh._.]

@@leosparton622 what u studying???

[@leosparton622]

@@Ansh._. automotive engineering aka mechanic

[@Ansh._.]

@@leosparton622 gotcha, don't worry bruh u'll be a brilliant automotive engineer 🗣️✨ my well wishes are with you

[@henryconner780]

This dude has spent thousands of hours helping students of all ages for free. Bro IS philanthropy

[@davidokoro2084]

THIS IS WONDERFUL. Now I understand the idea of limits

[@Vincent-qy4rq]

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

[@obisvanainobis9950]

I am happy to say that while I was absoloutley confused on what a limit was I finally had some idea on the topic by the end of the video

[@sinamqd]

Your first example was so good to introduce limit to a beginner.

[@zafrinrahman1091]

"Not the most beautifully drawn parabola in the history of drawing parabolas" 😂

[@rebeccabriggs1991]

I am definitely failing calculus. Fuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuck me.

[@aerick9463]

me rn

[@saqqurachan6887]

so what happened?

[@algeria3033]

Howd everything work out??

[@jessegabriel1438]

may i lol

[@powpowouchy5]

depends are you legal now

[@opallennoch655]

Super glad there are others to explain this!! While Math is a universal language I found it really hard to understand what my Professor was trying to convey with his heavy accent.

[@indigoGoddess7]

This makes a lot of sense to me. I hate how in high school the teachers made things seem so complicated. Thank you so much for all of your hard work. I really appreciate it. 😊😊 ✨✨✨

[@nickioli13]

One point nien nien nien

[@braisfernandez5687]

NickioliRavioli lul

[@JiaJiaing]

This is such a clear simple and concise explanation! I am currently learning calculus for the first time this helps a lot. Thanks!

[@adellagnoori]

Thank God for Khan Academy! You will save me in my college Calculus class! :)

[@huaiyuanjing3881]

It's simple but clearly and tells thre most important things.

[@toothfairy5352]

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

[@jaysplace01]

this is actually pretty sweet I'm thinking about going into mechanical engineering and there's a lot of calculus involved I'm definitely gonna watch the hell out of this channel.

[@swag6514]

from a mechanical engineering student, you better be ready lol and this channel will help you a lot too

[@jaysplace01]

the major is really kinda scaring me. I'm doing an AA in automotive technology and we go over some engineering through our courses but math and chem have always been my kryptonite...

[@EpicXXProductions]

Chemistry is easy if you put time into it and you like it. It's actually really interesting.

[@jaysplace01]

I've taken two years of it and I hate it with a burning passion. I've also had crap teachers who never helped me out with stuff I had difficulty with so that also didn't help

[@anthonyshort7265]

oh and these videos have already helped me pass 7 tests so thank you so much :) your a life saver

[@sahilnarang2986]

Thank You Sal, That's a great help !

[@footeythegreat]

what this whole video is saying is for any given function, the given limit is equal to whatever point the function is approaching. just saved you 11 min

[@dannycamello6106]

footeythegreat you need to show work

[@pikapuff123]

Yeah, but the insight is that you can do this even when you approach an x-value that isn't in the domain of the function.

[@z1xl423]

This is better than how my teacher would explain it

[@hankc6208]

My calc teacher didn’t even bother to teach me this. Just told the entire class to pull up this video. Only bums don’t actually teach their class.

[@JJ-zr1wf]

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

[@samiyaislam2530]

My professor never taught the basics. So I never felt the math and didn't understand a single class. After watching the video, it is so simple and interesting to me. Thank you Khan academy ❤

[@prmamidi]

i wish my college professors were like you! Thank you so much!

[@hankc6208]

Honestly ur calc professor can’t be as bad as my HS teacher. Mate this guys literally an egg white

[@awesomecake_abc2748]

how can i use this to find when my dad is coming home?

[@rajvardhanshinde6251]

I love you khan Academy, Every time I watch your videos, my basic concepts become more clear.

[@musix-freeze2452]

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

[@Notchworld]

Im taking calc now and its going to be hard. I love math even though i'm bad at it. My positive attitude and work ethic are reasons i do well. I admire math and i want to be the best i can be at it! I have to put in the time though!! I took statistics last semester and got an A and my teacher was phenominal. However, calculus is algebra based and that is my weaker area. My teacher is an older asian woman and we dont connect😐 i look around at the other students and they all have a pissed off look on there face because she doesnt teach well at all. Thank you khan academy for helping me and thank you to all the fellow youtubers who respect this subject.

[@The112Windows]

You can do it. With enough determination and motivation of course.

[@Notchworld]

i ended up getting a B

[@bradleynathanson2709]

TheRealDavieLondon lmao

[@DhrithionVocals]

ratio

[@jacklorenzrivera655]

Thank you! This is really helpful!

[@pantydropperr6002]

We finished the chapter today and i didn’t understand a thing in school so cheers to sal for basically putting me through high school

[@souvagyapaul8989]

Thank you so much for making the concept of 'limit' so simple as well as interesting for me🙏❤️

[@TheFallenJ]

you sir,seriously,can teach me in 10 minutes what my high teacher can't explain in one week

[@hankc6208]

So truuuuuu. Lucky for you my calc teacher is a egg white. Going to make fried rice this weekend❤❤

[@brianmutanya4537]

May God continue blessing you sir i understand alot of things

[@fahrirexhepi1065]

You make my life easier , thank you.

[@VeryNotSober]

You can graduate college amazingly just by watching his videos :>

[@nidulamaknidulamak4214]

I feel like my soul leaving my body

[@natalia4903]

thank you for being the first one to actually make this make sense

[@unknwn8696]

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

[@chochomunkeechochomunkee7483]

"not the most beautifully-draw parabola in the history of drawing parabolas"