Introduction to limits 2 | Limits | Precalculus | Khan Academy

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Introduction to the intuition behind limits
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Precalculus on Khan Academy: You may think that precalculus is simply the course you take before calculus. You would be right, of course, but that definition doesn't mean anything unless you have some knowledge of what calculus is. Let's keep it simple, shall we? Calculus is a conceptual framework which provides systematic techniques for solving problems. These problems are appropriately applicable to analytic geometry and algebra. Therefore....precalculus gives you the background for the mathematical concepts, problems, issues and techniques that appear in calculus, including trigonometry, functions, complex numbers, vectors, matrices, and others. There you have it ladies and gentlemen....an introduction to precalculus!
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Опубликовано:

28 сен 2024

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Комментарии : 473

@cheguevaraisgod 10 лет назад

890,000 views. The most attended first year Calc lecture to ever occur.

@gabe80million 6 лет назад

1,589,833 views now

@tanishbajaj9517 6 лет назад

1.6+ M now, as i have noted the + sign, dare beat me!

@garysgroundschool4739 5 лет назад

Like if in 2019 and taking calculus 😂

@saino5382 5 лет назад

1.7 Mill. in sept. 2k19! 😀

@umeshsinganamala9792 4 года назад

1,739,473 views now in November 2019 😂

@khanacademy 17 лет назад

A limit is a definite number. I'll do a video soon on the rigorous definition of a limit.

@arjundebnath2059 3 года назад

@kakalukium3337 3 года назад

Ok... Noice one

@thestudiousguy5252 2 года назад

Woah i would have failed my school if it wasn't for you sir

@souvagyapaul8989 2 года назад

Sir.. I can't explain in words my gratefulness towards you... Wish you will be continuing to guide us like this ❤️🙏

@ניין-י9ש 2 года назад

Wow 14 years ago.... Still helps You made history haha

@princepatel3427 Год назад

I'm watching this video in 2023. But there's nothing surprising that this video is still useful as it's made by most honorable person. Also watching this video inspired me how small things turn into invaluably precious things by making heartly efforts in right way. The way that Sal started deliberately on his small computer with incredibly low resolution inspires millions of common people worldwide to startup new things in their lives. Special thanks to the one and only real educator -'Sal'.

@fashionvella730 Год назад

me also

@iseeu3862 4 года назад

Those who are watching this after a lecture : RESPECT Those who are watching a day before exam: I can totally understand

@nathanielbeshane282 3 года назад

What about those who are watching before the lecture

@DrowQ 3 года назад

Me watching this and going to learn it in 2 years ._.

@nitinc7804 3 года назад

@@DrowQ you are a prodigy man

@ahaanhalwai6739 2 года назад

I am watching this cause I am curious about the topic of calculus

@saigeethaitv9832 2 года назад

You got me

@TheCaffeinatedDrive 10 лет назад

I'm calling holes "graps" for now on :D

@mehwishfatima1638 Год назад

Do you still call it graps?

@rampageblizzard 8 лет назад

for all of those whom don't get why lim f(2) is 4: My teacher always told me the limit is the number the graph is trying to reach. as you infinitely get closer to 2 on the x axis, the graph Is trying so hard to get to 4. due to the explained circumstances, the graph itself does not go through 4, however, that's what it's trying to reach.

@yashdave4752 8 лет назад

Thanks mate for ur kind explanation!!! Now I think I can define it in my own words..

@chandlerdouglas98 8 лет назад

*who

@DotaApoomUs 8 лет назад

i'm a visual learner and that helped a lot

@syafiqadly9428 6 лет назад

How about when lim f(3) ??3 on the x axis?can someone expains?

@AryanaSreshthaReede 6 лет назад

@@syafiqadly9428 9 as there are no obstruction mentioned like x does not equal to 3, therefore x^2 = 3^2 =9...I'm fairly new to this topic but that's how far I understood

@sugarandbones6272 5 лет назад

4:37 i just imagine sal coming home after a party and it's 3am and he just thinks hey, let's make a calculus video.

@TheLorelei 11 лет назад

You have the concept absolutely right. I think it is just the notation that is confusing, particularly since the notation uses an = sign. But what the notation means is as the graph approaches x=2 the y values will get closer and closer 4, but never reach it (until x=2). In other words, the limit is the number that the function is heading to as x gets closer to 2 (or whatever number we're interested in). I'm sure you've figured all this out by now, but maybe it can still be helpful to someone.

@mybabyalulu 6 лет назад

5 yrs later, very helpful to me. Thank you very much

@mohamad5762 Год назад

9 years later helpful to me

@tiffaduckie333 15 лет назад

I just want to say that Sal is an great teacher. I'm a college calc I student who is REALLY bad at math and I am pretty sure I'm going to pass my calc class thanks to this guy! (Final is tomorrow...wish me luck!)

@nullchinesh265 2 года назад

Did you pass?

@henok4121 Год назад

are you even alive bro ?, it has been 13 years

@inversechaos9256 Год назад

Bro you alive?

@mehwishfatima1638 Год назад

How's life going?

@jaabirana 14 лет назад

calculus is not hard but very time consuming.....i really appreciate your videos and i rate them 5/5 without a doubt cuz they help me so much....may god bless u men...

@mrpee2482 4 года назад

When this video was made I was at preschool .... But 12 years later I am still enjoying the works of Sal 🔥🔥🔥🔥🔥 Well Done

@miken5771 9 лет назад

Now 964,000 views. Let's push it to the 1,000,000 mark.

@pear201 9 лет назад

+Michael Nguyen 995,000 SO CLOSE

@EthanBrierleys 9 лет назад

+Michael Nguyen we did it

@spooky6703 8 лет назад

+Michael Nguyen I honestly wish the views started reading 999,999.9 recurring.

@fiervainqueur7818 7 лет назад

PUSH IT TO THE LIMIT! Scarface approved

@tanishbajaj9517 6 лет назад

lol i am reading this now in 2018

@neomillmarcreas6535 6 лет назад

Your voice was much better than the previous vid.

@SO-dl2pv 5 лет назад

The concept of limit is relatively simple but people find it very confusing, I will tell you why. Most of people think that functions have a single expression in other words they think that functions are only elementary functions (polynomials, trigonoemtric, exp,... Etc or combination of them). The concept of function itself is not well explained in books so people say why in the hell we have to invent the concept of limit at all if we have just to plug the number in the function and evaluate it. In short in mathematics the concept of function is more general than elementary functions. In most applications we are familiar with basic functions. I remember once when I saw a function defined as cases (multiple expressions) I thought that is not a function but mathematically it is a function.

@joymella123 15 лет назад

Thank You Man... YOU ROCK, I have been sitting in my cal one class for over a month, and did not understand what a limit was or how to evulate one until this video... YOU Rock! If I ever hit the lotto, I'll be sure to compensate you!!!

@sarbanisinha1425 6 лет назад

This man actually made studies a real gift for everyone

@ayagjian 8 лет назад

haha.. gap in the graph = grap

@oldsoul2829 6 лет назад

marvellous..now I am confident enough to take tution for highschool kids on limits

@haitiancreolewithluciano 3 года назад

*Loved the ‘grap’ joke*

@laplandJJ 15 лет назад

Great explanation..with a good sense of humor..thanks

@MsMusic816 12 лет назад

Btw great video! wish all teachers were as helpful as you!!!

@waterlily716 15 лет назад

oh, i already took cal 1 (actually it was AP Calculus AB). limits are like the first thing. :) what i meant was this whole series on calculus will be useful in cal 2. i hope it covers some of that stuff, but even if it doesn't this is still really great for review.

@36goldfinger 16 лет назад

i watched this video and went for my math exam .. it really helped .. good job man.. thanxxxxxxxxxxxxxxxxx

@aznlaughable 14 лет назад

My calc professor said to us(a class of about 50 students) the other day that there will only be ten of us who will get through the class and make it to the final. Apparently he has a reputation of being challenging. I'll be watching your video religiously. So far I've watched a couple of the trig video as a review and so far so good. I really like what you've done and hopefully these video will help me like it has for iMNice99163

@Sharmakee 13 лет назад

I am finally going back to college since math was my least favorite subject thanks to you Salman!!

@jasonli1060 8 лет назад

Do limits always have to be values off the y-axis? Just for verification, limits are numerical values that a function approaches but cannot equal to?

@abnerm23 12 лет назад

you should do a continuity video for calculus

@MsMusic816 12 лет назад

Omg! I wish I knew about Khanacademy a long time ago! It would of helped me out so much!! Dx

@DiLiG3N7 7 лет назад

Push it to the limit! This song just popped into my head lol

@rgh1986aka199 14 лет назад

@jeminijem1 No. This question has been in his faq thus.. What equipment do you use to make your videos? I use Camtasia Recorder ($200) + SmoothDraw3(Free) + a Wacom Bamboo Tablet ($80) on a PC. I used to use ScreenVideoRecorder($20) and Microsoft Paint (Free).

@aznkpride2005 15 лет назад

lol watching these are actually keeping me on track during winter break god damn i have no life xD

@wingsforfuture8147 Год назад

Thank uuu, it really helping meeeee

@interiorptr 14 лет назад

@hedonism Actually he might be going the hard way but it all connects till video 4. My stupid high school teachers practically used to tell us to solve limits by just substitution, then they just had us remember rules, "oh solve it like this if you see that" he explains WHY you simplify the function when you see it undefined and what you're really doing there. Very awesome!

@nandnijotwani7584 7 лет назад

Thanks that was helpful 😃

@MrRainar007 12 лет назад

is there a video on continuity sir ?i badly need only tht part... please notify me soon

@tammisettimuralikrishna8072 8 месяцев назад

Thank you so much sir ❤

@marvelhasiholan5495 8 лет назад

You sometimes sound like Jimmy Fallon, it's great!

@poshstars 15 лет назад

thank you its all clear.

@AnshumanGogoi7 14 лет назад

You dnt knw he much your videos have helpd me Sal. I m so glad i found you on youtube....^_^

@toancougar9239 11 лет назад

and yet he still teaches better than most

@MsGrammarnazi 15 лет назад

Awesome teaching skills.. Thanks for the video. Dana

13 лет назад

Well done !!!!

@gachakiritokun1102 4 года назад

What is a gap in a graph means? Please explain it?

@jeetukumarkv Год назад

thanks

@sabyabhoi8841 6 лет назад

Can a function have more than one limits? Like that of a tan function or other trigonometric functions where at certain angles, the function turns undefined.

@adrianapomatto1614 5 лет назад

Loving these videos!

@JoeVirella 14 лет назад

Awesome explaining and funny too. Thanks dude!!!

@sjung94 13 лет назад

@bodinian my teacher told us to learn it ourselves... we have quizzes everyday. ohh shooott

@sasukeuciha188 8 лет назад

Thanks for the videos. Sorry,what kind of tools or software that you used to make these videos? Thanks.

@fyaerwe 13 лет назад

I like your videos, but I'm still confused as to what a limit is :( I had to look up other videos, and one video explained that on the graph, it's a matter of the relationship between the vertical position and the horizontal position. Please don't assume that anything is obvious. It may be obvious to you but it may not be to others.

@zackjames2409 4 года назад

What does mean h Æ 0 under a limit? Is it same as saying h -> 0?

@Noodlepony 12 лет назад

This is the meaning of life

@kartikraju4101 4 года назад

Oct 16th,2020

@macmos1 13 лет назад

i didnt get the second part?

@kenan8935 7 лет назад

a grap? Subscribe

@tamagosan75 15 лет назад

I've been watching a lot of these, and many of them suffer from the same problem; trying to learn new concepts is extremely difficult when the person doing the presentation is making constant little asides to himself about the program he's using. At best it's a distraction, at worst it's actively annoying. A little investment on Sal's part in learning the whiteboard program he's using (or maybe getting a better one) would help tremendously.

@DelJuvePiero 12 лет назад

Promise me never to quit!!!!!!

@hedonism13 14 лет назад

I hate people who hate Khan Academy, but this time, I think this video is Khan's.... only let down. His science vids, and his (other) maths vids, they were all awesome, but this... I didn't get it until about 5 minutes ago, and it wasn't thanks to Khan. This is a really simple concept, and it seems like he's decided to go the hard way around for this video. But, if other people found it easier to learn it this way, ok.

@Lofeupsidedown 8 лет назад

its of gr8 use :)

@AlphaBurst 8 лет назад

So we're finding the value of the limit?

@mahadhashmi9348 2 года назад

What he said in the end about the limit of f(2) not being the limit of f(x), how does that work? Like limit of f(x) is 4 and 2^2 is also 4 so how is that not the same?

@maybeinactive Год назад

No, he defined the function as f(x) =/= 2 and if x = 2 then f(x) = 3 (which isn’t true cause of how the lines are drawn in the graph. The whole point was to show how using limits can help Alleviate some of the problems that might occur when we put different functions in a Cartesian plane) Adding the limit helped him find a value which is in close approximation of f(2)

@mastermax7777 11 лет назад

it approaches infinity

@suvendrajenamani5671 7 лет назад

Are all parabola of graphs of quadratic polynomials symmetrical?

@manaalla8507 Год назад

Super thnx,sir

@brideofabull 15 лет назад

grap!

@Peter_1986 11 лет назад

No, he sounds relaxed. =P

@devilliersgayle8231 6 лет назад

nice.

@christophermourao6293 10 лет назад

what program you use to did calculs ? program photoshop elements ?

@connorwinterspianotutorial9253 10 лет назад

I believe he now uses a program called SmoothDraw 3.

@devanshusachdev7367 Год назад

One Question- Why this concept? What's the use in real life?

@444zane3 13 лет назад

i was gonna say a science joke but this is a math video and all the good science jokes Argon

@facechallenges9 8 лет назад

very explicit. Thank you very much.

@LadyHelenaRavenclaw 12 лет назад

we have a grap LOLOLOLOL we love you

@irappapatil8621 4 месяца назад

In my opinion,limits are unwanted consequences.We can do better without limits.When Newton did it,why not we?.

@AkashYadavOriginal 12 лет назад

7 month before there were 22 now 5 more got worried

@hahs4 12 лет назад

yes, a GRAP

@aliyarae 14 лет назад

"...a gap in the graph, which could be called a Grap."

@PhamLindah 12 лет назад

took my maths teacher 2 hours to explain this limit thing and at the end of the lesson I didn't get a thing.

@fernandoraphael95 6 лет назад

graaaaap

@smitrib 12 лет назад

somebody has to put the REAL definition of "grap" on urbandictionary now HAHHA.

@72dew 13 лет назад

@suliemanhs what does a guy rapping have anything to do with limits, again?

@willdraper1141 5 лет назад

I'm pretty sure this is don lemon. not sal.

@XAttaHabibX 12 лет назад

my professor told me to come here.

@fogofchess 15 лет назад

function

@monicataco3734 6 лет назад

Nice to know I have 1.5 million buddies who also don't understand Calculus.

@frank-ts1yr 3 года назад

Ha, never stop learning im 44 and i dont know it

@theonewhocooks111 2 года назад

Im taking it this year and I hardly even know what its about

@wantsomecandies73 2 года назад

@@frank-ts1yr You gave me hope.

@sarju7166 2 года назад

😂😂

@shahnoorsameer4350 2 года назад

Hmmmm 5537238 iota

@rmc10141991 9 лет назад

this crap is a life saver, i don't learn shit in my actual class, but the combined with the text book and friends is the only way i can learn anything. And some how this actually makes it easy for me.

@shashankp 5 лет назад

Actually I do things the same way

@mfkrmaggot2755 Год назад

*this grap

@sabo2603 6 месяцев назад

Me who is 14 :)

@philv2529 8 лет назад

wouldn't it have been cool in school if our blackboards used neon colored markers. It makes for a better presentation. (yes i realize that this is not with markers but a comp program)

@NoraSeed_1508 2 года назад

We have a smartboard in our class and it looks almost similar to the video when something is written on it .......but the background looks dark(darkest?) green instead of black and it doesn't look good. LOL .......Honestly i feel there is something satisfying about the creak of chalk while writing that makes it special 😅

@1073601 14 лет назад

it's crazy how you've helped millions of students, including myself thanks

@Anish-IITP 3 года назад

And now the number became a billion

@unnamedchannelowouwu Год назад

It's crazy to find old comments, when people wasn't as fake as now and they showed their real opinion

@s.sharma8105 Год назад

👏

@trebelojaques458 2 года назад

_grap_ Haha love your efforts and amazing explanation. You've definitely changed the education system for real, one of the few golden drops here. ❣️

@airenmehnaz 2 года назад

lol I was about to comment that and saw your comment🤣

@marianesanchez9464 9 лет назад

omg a grap

@sophc193 9 лет назад

dude. thank you so much. i'm just starting calculus and was already freaking out (today was literally our first class and just review and an overview). now that i've watched this video it's calmed me down a lot; i feel ready for tomorrow's class. thank you! :)

@ariabeatrice6217 7 лет назад

Wow how industrious you are, to prepare for a class like that. I'm literally watching this bc I have midterms tomorrow and I don't know a thing.

@Hannahruda 3 года назад

6 years later and students (like myself) are in your exact same spot. I can only hope it went well for you and that it will go well for me also

@Arthur-fs5vl Год назад

How was High School and University or College for you? It's been 8 years since this comment, I was only 10 at the time.

@sophc193 Год назад

@@Arthur-fs5vl this is such a sweet comment! switched into the arts because calculus was not for me (as the comment 8 years ago implies), much happier now and making lots of art! hope it goes great for you too :-)

@Arthur-fs5vl Год назад

@@sophc193 Thank you very much Sophie, wish you the best in your future art projects and God bless!

@nejcalvoblanco 10 лет назад

Why does he sound so happy at the beginning?? XD

@jounynFM 10 лет назад

He loves Limits dude.

@Hemenmyname 10 лет назад

jounynFM funniest, best timed comment on youtube :D, it's totally viable he loves limits that much :D, it's Sal after all :D

@discuss2130 9 лет назад

jounynFM Too much. Not needed to understand calculus.

@TheWizardVampire 4 года назад

I think it's great that you keep showing students how the functions relate to the graphs instead of just showing them separately. It made limits feel a lot more intuitive to me. :)

@lizzyberg1082 4 года назад

I don't know if youtube pays you or not and because some of us cannot afford to donate we can definitely watch adds! please go ahead and add at least two adds in your videos to support fundraise!! Thanks, soo much from Pakistan!!

@blackpdf1239 10 лет назад

Thumbs up for those who see this videos day before the exam.

@mrslevitation 3 года назад

13 years ago😳...anyone

@MuhammedAshiq-f3c 4 месяца назад

16😂

@ryancochran3238 7 лет назад

What field uses these functions? I'm a draftsman for an engineering firm, and with all the math I use on a daily basis, I've never seen these functions used.

@sharkofjoy 5 лет назад

I don't know that any field uses these directly, but these are foundational to higher level maths that I do use in various types of computing.

@jimmysuros6302 5 лет назад

Chemical engineering.

@sammynnamdi4803 5 лет назад

Am also a draughtsman maybe we can just use the graphical approach and leave the analytical analysis for them lol

@sammynnamdi4803 5 лет назад

Trigonometry uses limits more than calculus but its not a field

@salmannizar7571 3 года назад

Bro, This video is one of the oldest videos we have. Its a treasure.

@gobberpooper 15 лет назад

thank you thank you thank you thank you thank you. I have ADD and although I loved how simple and easy to understand Calculus For Dummies was compared to my brother's college textbook, it's still a book and I quickly get bored and leaving. These videos are wonderful and I will be comming back for the whole series and perhaps for the physics and chemistry ones as well.

@gwa1654 9 лет назад

Im from Brazil and Im just starting calculus. I attend civil Engeneering and i am so exited and nervous and such fritened.

@gwa1654 8 лет назад

Actually, I had to start over. So, I am attending calculus 1 again. And this time I hope I'm gonna succeed.

@gwa1654 8 лет назад

+Sergen Yurdakul good luck for you dude! We'll gonna succeed. ;-)

@anony00000000 8 лет назад

+Guilheme Wallace According to my experience. If you stick too much to the proof of the basic equations and thinking why they're that way, instead of accepting them just like that and use them for calculation. It'll be hard to pass.

@gwa1654 8 лет назад

Ann Schreiber yes, you're right. I also think that we should only do math the way it's supposed to do, not asking why, why, why. But I think this time I'm going better, I've just took my second test and think I was successful. ;-)

@gwa1654 8 лет назад

***** sometimes I think it's like this. It's very hard to understand every details while we have a lot o things to learn and a lot of matters.