precise definition of the limit for multivariable functions (KristaKingMath)

► My Partial Derivatives course: www.kristakingmath.com/partia...
In this video we'll learn about the precise definition of the limit for multivariable functions, also known as the epsilon-delta definition of the limit.
● ● ● GET EXTRA HELP ● ● ●
If you could use some extra help with your math class, then check out Krista’s website // www.kristakingmath.com
● ● ● CONNECT WITH KRISTA ● ● ●
Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student-from basic middle school classes to advanced college calculus-figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: www.kristakingmath.com
FACEBOOK // / kristakingmath
TWITTER // / kristakingmath
INSTAGRAM // / kristakingmath
PINTEREST // / kristakingmath
GOOGLE+ // plus.google.com/+Integralcalc/
QUORA // www.quora.com/profile/Krista-...

[@hecz250]

When I turned the page and saw that dreaded epsilon symbol, I felt like throwing my chair out the window and shrieking like an ostrich in labor. But thanks to you, all is well. I really appreciate how you summarized the 3-part procedure and gave reasoning to your steps. Thank you *so much* for this detailed and digestible explanation of limits.

[@Blox117]

but do birds go into labor? i thought that was the whole point of laying eggs was to avoid giving birth like most mammals lol

[@osuh4947]

@@Blox117 Still, imagine yourself pooping out a giant egg. Wouldn't be that pleasant i think.

[@ahanshetty9034]

8 years on and still capable of clearing the doubts of any confused math student. Great video Krista!

[@AndreaLovelieee]

BEYOND AMAZING VIDEO!!! My professor NEVER shows step by step as you did. I am so thankful for professors / tutors like you that show STEP BY STEP for people like myself. Thank you for such a helpful and AMAZING video!!!

[@KartikayKaul]

Thank you so much! Honestly, college professors never help and do not give much time to explaining stuffs. Your video was able to explain this concept so well within about 30 minutes! The geometric interpretation also helped a lot in understanding! I love you! Thanks again!!!!

[@kristakingmath]

You're welcome! I'm glad it could help! :D

[@luigivincenzo9731]

That awkward moment when the example used in this video is exactly the question in my maths assignment...

[@dannygrim02]

Hahahahahahaha,u are the luckiest guy ever

[@milkyasleul3673]

I wish that awkward moment u talk about happens to me tomorrow in my test

[@khurramqasir6815]

That means your professor and Krista have something in common. Maybe they should date. 😂 😂 😂 😂

[@aarifhussain3700]

@@khurramqasir6815 I think so😂😂

[@mr.jaydeepmakwana7457]

The same problem asked in my assignment,😆

[@caducoelho2221]

Colorful boards videos rule. First KhanAcademy, and now I've found your channel. I'm so glad people like this really exist.

[@akshatgururani1149]

This is probably the best lecture I ever listened to ,for multivariable Calculus. Thank you so much.

[@kgetsepemkhabela3963]

My sister, you're the best. You simplify things much better. Thank you.

[@bassamaboona6444]

Thank you very much for this thorough explanation! I am taking calculus III online, and this video really helps understanding the concept.

[@RealationGames]

Brilliant video! I had to look 5 different clips on KZbin and this was the only one that explained this fully to me. Thanks!

[@kristakingmath]

you're welcome, i'm so glad it helped!

[@oguzhandogan4947]

so glad i found you. thank you for explaining these complicated things articulately. and i must say you have a beautiful voice :D

[@alphastrategygroup7422]

Great straight forward explanation. Kudos. A mind well thought of is divine.

[@silversky216]

I can't believe this video was posted 7+ years ago. Absolute quality!!!

[@jamesd1366]

Really great video! I really appreciate how your explanation is so detailed and how you started with explaining the concept in 2D before going to 3D!

[@kristakingmath]

Glad it could help! :D

[@thetruereality2]

Thank you very much. Very much grateful for devoting 35 mins of your time for this.

[@elainem8591]

Thank you so much! You're an amazing teacher.

[@AnaximandroAndrade]

YOu are awesome, the best explanation of limits I ever saw!!!...Thanks!

[@thepotterheadful]

Thank you for explaining everything so visually!

[@lakshya6235]

awessome ,surely saved my times to go around several books .

[@paulj.murphy7447]

Wow. Excellent, Krista.

[@efrenpandesandmeihlzen9855]

Great,,,very well explained. Thanks for the video

[@Skhulile84]

Wonderful, keep up the good work!

[@eddierosell9329]

Wow! I'm studying engineering physics in Sweden and I've been reading about the epsilon, delta definition but never quite grasped it completely.. You just made me understand it so much better than my books ever did. Thank you!

[@kristakingmath]

You're welcome, I'm so glad it's finally making sense! Such a tough topic to explain and to learn! :)

[@MrJon0813]

Quite a long video but very worthwhile to watch. If I can like this video a million times I would.

[@estebanm8929]

I have to say. The world needs more teachers like you. Because human kind is very capable but we are misguided.

[@Hemenmyname]

Great lecture, neat, rich and clear. Thanks!

[@kristakingmath]

+Sirius's Apparition You're welcome, I'm glad you enjoyed it!

[@Jono.]

This was so unbelievably clear. Thank you!

[@kristakingmath]

:D

[@alkankondo89]

Thank you for the well-explained and thorough lesson over these proofs. I hope to eventually get good at writing Calculus proofs like this, and this video was a good place to start!

[@kristakingmath]

+alkankondo89 Glad you enjoyed it!

[@pedromoro561]

Congratulations for the videos: superb and clear explanation. Looking forward to see another topics in calculus and mathematics in general.

[@kristakingmath]

Thank you so much, Pedro! I'm so glad to know that you like the videos! :)

[@MrALCHEMIST2010]

You presented quite neatly and easily understandable way. Great job !

[@kristakingmath]

Thanks!

[@mongoqueen123]

Wow! I tried to understand the epsilon-delta definition like for half a year and i really didnt get it. Your awesome explanation finally made me understand it and with the example i also know how to use it now ! Thank you so much ! You are a awesome teacher! Saludos

[@kristakingmath]

Yay! That's awesome. Thanks for letting me know!

[@hamisiwenge1369]

this helpful thn 1 hr lecture thnks a lot

[@calvinlee3652]

Can you give any examples of when you tested various paths and got the same limit, but when trying to prove the limit exists with delta-epsilon, it doesn't work because the limit doesn't exist?

[@mausefue]

Thank you so much, your videos are so helpful.

[@LapizDragon]

This video, and the one preceding it were perfect. Not only were they the exact same problems as the first two problems of my assignment, they were really well explained. I wonder how well I'd get this stuff if I actually made it to lecture.... Thanks a ton!

[@kristakingmath]

+Dean Arnesen You're welcome, I'm so glad they helped!

[@omarkadry4143]

very great work .. very helpful .. best lecture video in this topic :) thank you too much keep it on :)

[@kristakingmath]

+Omar Kadry Thanks!

[@trackmyactivity]

+CalculusExpert.com Great, thanks :) During the lecture in 22:10, you use the word "discontinuity" and "indeterminate form" interchangeably. I think you actually mean "indeterminate form" (in the first step, when we plug in the values and get 0/0). (Having discontinuity would mean that we can abort the process of finding the limits, since there is none.) Or did I miss some concept? Thanks for the great lecture :)

[@bufferly5595]

Excellent video, you've blown my lecturer out of the water!

[@kristakingmath]

+Bufferly Thanks, I'm glad it helped!

[@LariosGiveNoFucks]

Holy crap, my mind was blown within the first 17 minutes. You explained it far better than both my Calc 1 and 3 professors ever did.

[@kristakingmath]

😊

[@harshitsinghai1395]

No doubt , you're a king Thanks a lot. You really saved my day.

[@kristakingmath]

I'm so glad it could help!

[@SebastianLopez-nh1rr]

Excellent video, congratulations.

[@kristakingmath]

+Sebastián López Thank you very much!

[@Bartolitz]

I love your videos!

[@jeremymel3779]

Great explenation!

[@amitrajo]

Amazing video, I have question if x^2+y^2 < r^2 , ABS((2x^4y+y^5-5x^2y^2)/((x^2+y^2)^2)) < b and b> 0, find the value of r?

[@proxy8918]

Amazing explanation, thank you!

[@kristakingmath]

You're welcome, Raphael! I'm so glad you liked it! :)

[@conorbanville9414]

Hi, I'm wondering when choosing paths to test along, such as x=0 or y=mx does the fact that we are approaching the point (0,0) factor in. For example if the question was to prove the limit D.N.E when (x,y) -> (1,2) would we choose different paths? If anyone could answer this I would really appreciate it. Thanks

[@pratyushsarangi1996]

nicely explained but i have a doubt can we derive a conclustion that a sharp change in the normal's inclination of a surface over a short interval can be meant that a function is discontinous and does it imply that the projected area of that curve from its highest value to its lowest value will give a perfect radius

[@IaKhanic]

Amazing video, but how did we assume that epsilon is the same as delta?

[@RoniJonathanBenKeuru]

Percise definiton of limit is formally said: For every epsilon greater than zero there exist a delta greater than zero so that if the radius of the circle on (x, y) plane is smaller than delta, then distance of the value of the function f(x, y) from value of the limit L is smaller than epsilon. So because Krista found a connection between epsilon and delta by showing the radius of the circle on the plane to be greater than the distance between function f(x, y) and the value of L, she was able to make epsilon and delta equally small and prove that the limit exists and its value is L = 0 while (x, y) goes to (0, 0). The main idea is that no matter how arbitrarily small delta we would choose, there would always be found values of the function between epsilon +/- L and L.

[@xaviergonzalez5652]

good!

[@vibodhj349]

@@RoniJonathanBenKeuru But I don't understand how can you make delta and epsilon equal to each other!

[@user-lp7mk3wn5s]

Im pretty sure she "chose" epsilon from the big inequation at the bottom right corner. [|x||y|]/sqrt(x^2+y^2) less than or equal to |y| less than or equal to sqrt(y^2) less than or equal to sqrt(x^2+y^2) It follows from that inequation, combined with the respective terms' meaning regarding the precise definition of a limit ( leftmost part is

[@rileymurdoch8810]

@@vibodhj349 probably a bit late to be replying to this (haha), but the definition says; for ALL epsilon, there exists A delta. Where it says the exists a delta, it literally just means there is ONE number (literally any number) for which this is true. Since delta can be any number, this means we are allowed to choose it. For this example in particular, it just happened to work well to say epsilon=delta. For other examples, it may not be the case that you want to make delta equal epsilon, you may want to make delta equal 2*epsilon, or epsilon/3, etc.. Basically, you can decide what delta is equal to, and you make this decision based on what will work for your proof.

[@josephevans1490]

Thanks for posting this video, it helped a bit, but I still have some questions. So the precise definition of a limit does not prove the limit itself, but only the fact that there is a limit? And therefore, because it was proven that there was a limit for the equation, the potential limit found earlier was proven to be true thanks to the precise definition? Furthermore(a bit of an abstract thought), so by themselves, approaching a limit from multiple paths cannot prove that a limit exists, and using the precise definition cannot prove that a limit does not exist? Or is that only true for when approaching through separate paths, and you could you prove and that a limit does exist using the precise definition only?

[@Thefakisgoinon]

I have two questions if you don’t mind. Can we not get rid of absolute value by simply writing -epsilon

[@abhishekhrit6922]

Very nice explained..

[@LouieMcConnell26]

Question: When would you have a case where the y = mx test works, but the limit fails when you test it with the quadratic tests?

[@erickgudin]

This is so useful and clear thank you

[@kristakingmath]

You're welcome, Erick, I'm so glad it helped! :D

[@seanbentley4459]

You're a bona fide saviour. A humble thank you.

[@aryansingh-ob2jd]

thank you so much. Really you are an amazing teacher.

[@kristakingmath]

+aryan rajput Thank you!

[@yugrawal5541]

thanks.. a lot.. it helped me in my engineering exams :-) keep it up... :-)

[@kristakingmath]

+Yug Rawal Awesome! I'm so glad I could help!

[@user-xh5ct4yw3x]

it is important to remember that the presence of an indeterminate form does not necessarily imply a lack of continuity. Continuity relies on the overall behavior of the function, while indeterminate forms require additional mathematical tools to evaluate the limit accurately.

[@Fazal828]

im scared to even begin watching this. Oh well no other choice

[@metawhirl4609]

Me too

[@sanjaysg6811]

Akka (sister in Kannada language) great video, u r my source of inspiration, please do more of these videos.

[@julianjawadahmad1006]

It's just!! WOW!! So good!! My concept gets strong base very easily. Thank you, a lot. Best of luck for your future videos (y) :)

[@kristakingmath]

+julian jawad ahmad Thank you very much, I'm so glad you liked it!

[@julianjawadahmad1006]

I hope you continue to make such nice vidoes future. Good luck, mam :)

[@TROLOLOLOLO007]

God bless. Thanks to you I feel confident on my next test. Thank you B)

[@kristakingmath]

I hope you rock it! :)

[@jaime7579]

Thank you very much. Really usefull.

[@stephberrie]

amazing!!! thank you! i finally get it!!

[@kristakingmath]

+stephberri Awesome! I'm so glad it makes sense now!

[@hamisiwenge1369]

ts btter thn 1 hr lectuer very helpful thnks a lot

[@denny8360]

near the 27min mark, the 0< was removed as the sqrt(x) must be positive, but can we rule out x=y=0 here? This video was very helpful! Thank you!

[@indrajeetchauhan5529]

Thank you very much! I have a clearer understanding of the topic after watching your video. However, I didn't quite get the last part where you set epsilon equal to delta. If we can arbitrarily set epsilon equal to delta, then why do we have to show that the expression that is less than epsilon is less than or equal to the expression that is less than delta?

[@Anthony-db7ou]

I hope you figured it out. That would mean there is hope for me :(

[@dankplatypus420]

You can choose epsilon equal to delta, because if [the expression less than delta] is less than or equal to [the expression less than epsilon], you can choose [the expression less than delta] is equal to [the expression less than delta] . So a=b , and a < epsilon, and b < delta, so choosing epsilon = delta would still hold true.

[@swe1337swe]

When u're setting xy/(sqrt(x^2+y^2)

[@MeganMcgahren]

Incredibly helpful.

[@kristakingmath]

Awesome, so glad!

[@aarifhussain3700]

Wao I really amagzed by your concepts so nice thanks very much

[@kristakingmath]

You're welcome, Aarif, I'm glad the videos are helping! :)

[@hamzahaif1129]

when i read Calculus book i was confused but now after watching this video i really understand how to get the limits of multivariable function

[@kristakingmath]

Oh good! I'm so glad it makes sense now! :)

[@matthewmuriithi306]

To be honest this is the first video I've seen of hers. I took one look at what's being shown, I heard her voice, within 5 seconds I subscribed.

[@kristakingmath]

Thanks for subbing, Matthew! :D

[@Bu7MaiD075]

2hrs of a Uni Lecture explained in 30 minutes. ALSO GOT TO THE POINT. EVERYONE UNDERSTOOD. DID NOT HAVE TO USE 20 LECTURE SLIDES. thanks yo. love ur explainations

[@kristakingmath]

Thank you! :D

[@sfxyz3746]

Such a good explaination

[@kristakingmath]

Thank you so much, Owen! :)

[@justinorji6479]

What if you insert "mx" for "y" and you end up with only "m" and constants left? In my question, I was left with (1 - m)/(1 + m^2). Does this count as a different answer?

[@juanverney]

thank you for the help.

[@chusrangagasirmarak4247]

This really helpd me a lot, m also scared of when i see epsilon n delta in limit but seeing this video m no longer scared of them :)

[@kristakingmath]

Oh good! I'm so glad this took the fear out of it! :)

[@oneday3611]

Thank u so much but why if we pose y=t and x=t the limit of f(x,x) = 1/2 its différent of "0" so the limit no exist !! contradiction we need to give diffrent values to X and Y right ?

[@c3realpt]

Thank you!

[@vrendus522]

Notation single and multivariable calculus.Note the straight line as op[posed to the parabola starting off from source.Since this is multivariable calculus,"reading this correctly in your intro"?Then a parabola does not really have an assigned value unless it is requested to do so. What I am getting is that as with some buildings that are work places, there is just a plug in that serves no real function, other than to be noted for its potential utility. Is this why the parabola is not assigned values of length and arc in you vid?This is what I'm reading am I wrong? Thank you

[@anasqureshi4606]

Hey Mam loved your voice as well as your explanation

[@VishalGupta-lw9dk]

Great video

[@MrDeyvid34]

Amazing, thank you very much

[@kristakingmath]

You're welcome, I'm glad I was able to help! :)

[@MarkNealJr]

At 28:00 can we use IyI instead of IxI?

[@Augustine_354]

Two words: Thank you :*

[@da-eunyim8070]

Thank you so much for the video

[@kristakingmath]

+임다은 You're so welcome! Glad you liked it!

[@tonicopm]

When you take x^2 out of the square root (min, 18:52) we do NOT get "just x", but in fact |x| ...

[@divneet3474]

if you stop uploading i will fail this class!!!! please keep uploading my future is in your hands!!! best math teacher evaaa!!!

[@kristakingmath]

Aww, thanks Divneet! I'm sorry I haven't been uploading as much lately, I've been working on content and projects for my website, but I'm hoping to get back to uploading soon! I hope your class is going okay! ❤

[@divneet3474]

@@kristakingmath major fangirling!!!!!! I m so glad ure making a website 📢📢📢!!! Never mind ,I'm reading from the book but the way u give an explanation really helps me, not everyone can explain things physically or graphically. My teachers just writes the formulas and some values and keeps moving forward!!!!

[@Cooper_sky]

We could use polar coordinates(r,theta) to approach a point in any direction, rather than choosing a specific direction such as lines and parabolas.

[@Anthony-db7ou]

Those moments when her voice changes, and you realize she's back from a break, lol. Great video. I honestly felt like I got it, and then the video ended and I did not understand how we proved the limit existed. I do not like using epsilon and delta, but my professor says there will be cases where we wont be able to use squeeze theorem... pray for me...

[@mustafaRoya]

Thank you professor

[@kristakingmath]

+Gholam Mustafa Ali You're welcome!

[@abdurrahimemin6483]

thank you so much

[@sebastianramirezcaseres2965]

Thanks a lot 🌻 Youre amazing and have a beautifull voice :)

[@kristakingmath]

Thank you so much, I'm glad I was able to help! :)

[@haleysmith5263]

I am passing Calculus 4 because of you. Thank you so much

[@kristakingmath]

You're welcome, Haley! I'm so glad the videos are helping! :D

[@AnuragPradhan101]

I love you ! You made my day ,Thank you very much for making this video ! I dont know how can I thank you ! Thank you word is very small to express my happiness !

[@kristakingmath]

You're so welcome!

[@sirazshaikh6818]

thanks great explanation keep it on :-)

[@kristakingmath]

+Siraz Shaikh Thanks!

[@iitmandi8048]

Thank you so much

[@kristakingmath]

You're welcome, Vikas! :D

[@Icecube88]

i'm glad our teacher didn't make us do this when i took calc 3. she only wanted us to show that the limit dne.

[@drizzy8450]

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303820

[@andreas_gswerve]

Can't get why we have to use the theorem to prove. If it's possible to change to polar isn't it better than getting confused with all that inequalities? In most examples I've seen u get either 0 meaning limit exists or some function with θ and because it varies u say the limit does not exist. Is that enough to prove not continuity? I'm too confused

[@heyjoe5048]

Thanks!

[@kristakingmath]

You bet, Jose! :)

[@TheiLame]

hehe going to watch this whole video tomorrow but i havent heard in ages that 0 is also pronounced ''oh'' - it kinda cut me off guard for few seconds. I thought it was mostly used with phone numbers. Im so used with zero being called zero. but basicly the L (oh) can be whatever value and not oh (zero)? little unsure