The only way to teach effectively is like a broken record. He never leaves anyone in the dust. Its nearly impossible to walk away from this class feeling lost or behind. So awesome
Intro to multivariable limits: 0:00 Showing a limit (with two variables) does not exist: 26:54 Showing a limit (with three variables) does not exist: 1:12:19 Showing a limit does exist: 1:32:07 changing to polar coordinates: 1:32:28 Continuity (domain and compositions): 1:41:47 Summary of computing limits: 2:06:56 - 2:07:24 Squeeze theorem (converting to inequalities): 2:07:26
Every professor should take note the beginning portion of your lectures. Every math professor I've had jumps straight to the computational stuff and expects students to learn what they're actually doing on their own.
One thing that bothers me is how in order to be a professor you do not need a degree in education. In other words you can be an expert but not know how to teach.
Hello Professor Leonard, I just want to personally thank you here at TH-cam for providing quality tutorials from starters to advanced topics at Mathematics. I'm currently at your Pre-Algebra courses, its just amazing! I've never been taught Mathematics this way which is very fun and intuitive way. Thank you so much!
Bless your soul Prof. Leonard, My professor explains things very strangely, using abstract notation, and she still uses blackboards without erasing the pre-existing chalk properly. You are literally going to save my life in this class and I feel like I am a part of the environment. You do a great job of explaining everything and it's worth watching you lecture for a little longer to make sure that I understand the concepts completely. Thanks for the humor!
these video might just save my semester. His explanations are so simple to follow and very clear and when my professor explains the same material I don't understand a single thing. Wish we had more professor like him. Thank you for the videos!
I refer many of my students to these videos, they are fantastic. With this one, I have one comment - around minute 43, part a says to be certain your point (a,b) is on your path. But then goes into testing along the x and y axis. If your point of interest is NOT the origin, do not try x=0 or y=0 as your path. Try x=a and y=b instead!
Thank You Professor Leonard! Your videos have helped me to Ace calculus I, II and III. I tell everyone who struggles with Calculus about your videos. They have been so helpful!
I can't believe how helpful are those videos. I'll watch them in class instead of listening to my professor b/c it's such a waste of time... thank you so much!
Went all around the internet looking for engaging and relevant content like this. This is so much better than my stanford math prof. I really appreciate your delivery of concepts (and this is the first youtube comment i have been compelled to make in years)
1:08:45 AND THAT WHY HE IS THE G.O.A.T!!! THE SMALL DETAILS, I love it! Thanks again Professor Leonard. Just killed my Ch.12 exam with a 99. Plan on going for a killing spree in calc 3.
Thank you for showing how to apply Squeeze Theorem to these problems!!! My teacher tried to explain it to us in half of a lecture and you did it under 10 minutes in a way that I can understand!
0:00 - Introduction to limits and continuity of multivariable functions 0:59 - Review of limits from single variable calculus 6:23 - Limits of multivariable functions 19:05 - How to prove a limit doesn't exist 24:08 - Example 1 (limit of function with two independent variables) (showing the limit doesn't exist going along the x-axis and the y-axis) 44:14 - Example 2 (limit of function with two independent variables) (showing the limit doesn't exist going along the x-axis, the y-axis, and a linear function) 54:17 - Example 3 (limit of function with two independent variables) (showing the limit doesn't exist going along the x-axis, the y-axis, and a cubic-root function) 1:02:30 - Example 4 (limit of function with two independent variables) (showing the limit doesn't exist going along different paths) 1:12:19 - Limits of functions with three independent variables 1:14:39 - Example 5 (limit of function with three independent variables) (showing the limit does't exist going along different paths) 1:20:51 - Example 6 (limit of function with three independent variables) (reparameterizing the limit: assigning "t" expressions to the independent variables) 1:24:18 - Example 7 (limit of function with two independent variables) (plugging in values to get direct answer) 1:27:22 - Example 8 (limit of function with two independent variables) (plugging in values to get direct answer) 1:29:19 - Example 9 (limit of function with three independent variables) (plugging in values to get direct answer) 1:31:54 - Example 10 (limit of function with two independent variables) (utilizing polar identities to evaluate limit) 1:37:22 - Example 11 (limit of function with two independent variables) (using polar identities and L'Hopital's Rule to evaluate limit) 1:41:48 - Introduction to Continuity in Multivariable Calculus 1:43:53 - Definition of Continuity in Multivariable Calculus 1:49:22 - Example 12 (continuity of function with two independent variables) 1:50:26 - Example 13 (continuity of function with two independent variables) (determining domain of function) 1:51:35 - Example 14 (continuity of function with two independent variables) (determining domain of function) 1:53:33 - Example 15 (continuity of function with three independent variables) (determining domain of function) (domain includes all points except the ones that form that particular SPHERE) 1:55:47 - Introduction to composition problems doing example 16 1:58:40 - Example 17 (composition problem setting "t" equal to f(x,y)) (domain restrictions with g(t)) 2:01:55 - Example 18 (composition problem) (domain restrictions with f(x,y)) 2:04:59 - Introduction to Squeeze Theorem 2:06:30 - Example 19 (Squeeze Theorem)
It's really a blessing to be taught by u. Your dedication level soars the sky. Thank you sir for making mathematics easy for everyone one of us! With love From India!!
My professor for Calc 3 is named Hongwei and speaks with a hard accept from china so when he goes over anything its always a bigger challenge to not only understand the concept but him as well. So when he went over this section he sped through it only talking about the epsilon-delta definition of it and then using polar to solve some problems, never fully explaining the actual concept. This video has not only saved my grade but also my understanding of limits for all future calculations I'll be doing in my career as an engineer. Thank you so much Professor Leonard!
It's funny how on calc 1, I watched your videos in regular speed. In calc 2 it was in 2x it's speed, and now I'm able to manually watch calc 3 in 3x it's speed. Just goes to show how clear it is, thanks so much for creating smooth transitions to new topics and info (:
As always, this is the best place on TH-cam for learning mathematics! Had some trouble at the end of the video though, and wanted to put a fact out there that tied the last section of the video up well for me: \lim_{(x,y) ightarrow (a,b)} f(x,y) = 0 \text{ iff } \lim_{(x,y) ightarrow (a,b)} \vert f(x,y) \vert = 0 Its LaTeX code so for anyone who cant decipher it just paste it in Overleaf or some other LaTeX editor.
Преп просто охеренный! Зачет. Cool teacher! I really enjoy his attitude. He is genuinely interested in what he is doing. On the other hand majority of university teachers barely tolerate students and see them as a nonsense and a distraction from scientific research.
In this first and second example if you do a 3D plot (Used Mathematica in my case) you can see that these functions are not continuous at (0,0) but has a cusp along one axis at the point (0,0) in example 1 and a cusp along the -x,y and x.y diagonals in example 2 therefore the limit is undefined. In the example lim as (x,y)->(1,0) of (2xy-2y)/(x^2 +y^2 -2x +1)Professor Leonard found that the lim as y->0 of 2y^2/2y^2 =0 presumably because this reduced to the lim y->0 of 2y^2/2y^2 is 0/0 =1 and concluded that the limit was undefined. However Mathematica finds this limit of the base limite to be 0, so the original problem has a limit of 0 also which is also what Mathematica found. This is in agreement with the plot of the surface and examining the point (x,y) = (1,0) Mathematica's determination of the original limit to be 0, so I believe his answer is differant. I guess this depends on your definition of 0/0 or it could be a bug in Mathematica. Please comment Professor Leonard and others if you can. Time point 1:11:48.
Your students are so lucky! Thank you so much professor this means a lot for a lot of us Your lectures are well delivered and clears out every confusion in my mind
I don't know if it's really good enough to say just a "Thank You"...but still "Thank you very much"....ur detailed conceptual explanations are a treat to watch and understand.
Professor Leonard, thank you for another well organized video/lecture on Limits and Continuity on Multivariable Functions with the classic Squeeze Theorem in Calculus Three. These concepts are introduced in Calculus One, which improve my understanding of the subject in Multivariable Calculus.
Following in the footsteps of SQ Flyer who got an A+ in math2011 intro to mul cal in 2019 fall, I have watched so many of your vids and I am going to be the next beneficiary of you! I managed to get a past exam question correct on 1/12/2022 but my semester starts on 2/4/2022. Thanks a lot!
Im from Argentina and this is the best video i found to understand limits, ty man the only one who say that the point have to be on the path, sorry for my english.
YOU ARE AMAZING!!!!!! I HAVE CALC 2 EXAMS COMING UP IN FIVE DAYS AND YOU ARE MY SAVIOUR TO ENDING THIS COURSE WITH AN A..:)))) thanks keep this up, much better than patrick jmt
Professor Leonard helped me through calc 1 with an F then round two I received a B-, then calc 2, I received a C and now Im in Calc 3 and who knows if I will pass or not! What Im telling you that just because you watch his videos and work your ass off does not mean you will automatically get an A or even pass. But don't give up people math is hard!
reviewing again, cuz i want it to be like...hate adding... like geometry when i was young. honestly, this is where it gets exciting. also, each time i study a lesson, as i understand it even more clearly, i realize even more what a good teacher you are.
Holy shit. Literally. Those kids learning at his class must realize how blessed they are being taught by a 'true' teacher (professor). I find myself in despair watching this video at the last minute just before my midterm test. Damn. Should have known him earlier...
Professor Leonard, I'm always excited to see new material. I'm currently in Calc 3, and you've been a major help along the way! I have finals next week and I can understand your probably very busy yourself... Do you plan to release more videos in the next week or so? Thanks!
+sandile mabika He actually already has released it, but the video was shot with a 4-D device in 4-D setting so unfortunately 3-D people are unable to see it. Maybe you should get 4-D glasses, I am selling if you are interested.
@1:36:45 someone asked why r approaches 0 from the positive side. r is the radius of the parameterized curve, radius is a distance , so always positive and that's why approaching from the positive side ✌
This video is a prime example of why I never go to lectures. TH-cam can explain it 10x better in a shorter period of time. Thankfully for me, medical schools have optional lectures.
My professor didn’t e explain anything, sadly. I have an exam tonight due at 12am, and I’m here trying to understand. Now I’m stuck trying to understand everything. Thank you for helping and existing though.
Thank you so much! I'm surprised my book never mentions the squeeze theorem and instead goes through a bunch of other non-sensical stuff... You made it all perfectly clear, and the theorem kind of makes sense intuitively as well.
i think you might be wrong here. When you use for example polar coordinates and you get a number it's not strong enough to prove that the limit exists, you'd have to use that number as a candidate of a posible value of the limit and then plug it the squeeze theorem or the epsilon-delta definition. Then just then you can prove the limit exists.
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Doing your job right. A simple thing. But oh my God, if everyone did their job the way this man does his, our world would be a far better place.
Absolutely I agree with you brother
absolutely
I think the world can be a better place even if we are not spoon fed
@@muhammetakman8085 I know
That is so true !
The only way to teach effectively is like a broken record. He never leaves anyone in the dust. Its nearly impossible to walk away from this class feeling lost or behind. So awesome
broken record tells how to understand others ,,,,,
Your username is so funny because Clark Kent is exactly who this professor reminds me of. :D
I see he's leaning into it too :D
@@emanuellandeholm5657 lool, you should see his pfp XD
love how you put this
Intro to multivariable limits: 0:00
Showing a limit (with two variables) does not exist: 26:54
Showing a limit (with three variables) does not exist: 1:12:19
Showing a limit does exist: 1:32:07
changing to polar coordinates: 1:32:28
Continuity (domain and compositions): 1:41:47
Summary of computing limits: 2:06:56 - 2:07:24
Squeeze theorem (converting to inequalities): 2:07:26
♈🔧
Where’s the balloon
Thanks a lot for the timestamps!
If I had this professor I would never miss a lecture. I firmly believe in a full understanding of the subject at hand. This guy knows how to deliver.
Every professor should take note the beginning portion of your lectures. Every math professor I've had jumps straight to the computational stuff and expects students to learn what they're actually doing on their own.
100%
That single introduction makes the difference!!! >:(
One thing that bothers me is how in order to be a professor you do not need a degree in education. In other words you can be an expert but not know how to teach.
conceptual talk until 00:00
Examples of 2 dependent variables: 24:00
Examples of 3 dependent variables: 1:12:15
Squeeze theorem : 2:07:25
Independent*
thanks
Elite
Thank you! I was just lookin for squeeze theorem
Hello Professor Leonard, I just want to personally thank you here at TH-cam for providing quality tutorials from starters to advanced topics at Mathematics. I'm currently at your Pre-Algebra courses, its just amazing! I've never been taught Mathematics this way which is very fun and intuitive way. Thank you so much!
ok so he is ACTUALLY explaining ive never had a teacher that did that im shook
Lmao xD
Bless your soul Prof. Leonard,
My professor explains things very strangely, using abstract notation, and she still uses blackboards without erasing the pre-existing chalk properly. You are literally going to save my life in this class and I feel like I am a part of the environment. You do a great job of explaining everything and it's worth watching you lecture for a little longer to make sure that I understand the concepts completely. Thanks for the humor!
Prof Leonard,
You got me through Calc 1 and Calc 2, both with an A+.
Thanks for your dedication to teaching!
I wish I had found this channel earlier- would have made a lot of things easier.
Same thoughts
these video might just save my semester. His explanations are so simple to follow and very clear and when my professor explains the same material I don't understand a single thing. Wish we had more professor like him. Thank you for the videos!
I refer many of my students to these videos, they are fantastic. With this one, I have one comment - around minute 43, part a says to be certain your point (a,b) is on your path. But then goes into testing along the x and y axis. If your point of interest is NOT the origin, do not try x=0 or y=0 as your path. Try x=a and y=b instead!
I am so grateful I came across this lecture. If every math professor taught concepts like this, the world would be a better place! Thank you so much!
There is no other way to describe this man other than a living legend
I love the way he teaches and his sometimes cringe dad jokes. Keep up the great work, I appreciate your help!
Thank You Professor Leonard! Your videos have helped me to Ace calculus I, II and III. I tell everyone who struggles with Calculus about your videos. They have been so helpful!
I can't believe how helpful are those videos. I'll watch them in class instead of listening to my professor b/c it's such a waste of time... thank you so much!
Went all around the internet looking for engaging and relevant content like this. This is so much better than my stanford math prof. I really appreciate your delivery of concepts (and this is the first youtube comment i have been compelled to make in years)
Thanks so much!! I'm very glad you are enjoying the videos :)
1:08:45 AND THAT WHY HE IS THE G.O.A.T!!! THE SMALL DETAILS, I love it! Thanks again Professor Leonard. Just killed my Ch.12 exam with a 99. Plan on going for a killing spree in calc 3.
I'm watching this an hour before my midterm and I'm already learning more than my teacher ever taught me
howd it go
watching this 8 hours before the exam
Thank you for showing how to apply Squeeze Theorem to these problems!!! My teacher tried to explain it to us in half of a lecture and you did it under 10 minutes in a way that I can understand!
0:00 - Introduction to limits and continuity of multivariable functions
0:59 - Review of limits from single variable calculus
6:23 - Limits of multivariable functions
19:05 - How to prove a limit doesn't exist
24:08 - Example 1 (limit of function with two independent variables) (showing the limit doesn't exist going along the x-axis and the y-axis)
44:14 - Example 2 (limit of function with two independent variables) (showing the limit doesn't exist going along the x-axis, the y-axis, and a linear function)
54:17 - Example 3 (limit of function with two independent variables) (showing the limit doesn't exist going along the x-axis, the y-axis, and a cubic-root function)
1:02:30 - Example 4 (limit of function with two independent variables) (showing the limit doesn't exist going along different paths)
1:12:19 - Limits of functions with three independent variables
1:14:39 - Example 5 (limit of function with three independent variables) (showing the limit does't exist going along different paths)
1:20:51 - Example 6 (limit of function with three independent variables) (reparameterizing the limit: assigning "t" expressions to the independent variables)
1:24:18 - Example 7 (limit of function with two independent variables) (plugging in values to get direct answer)
1:27:22 - Example 8 (limit of function with two independent variables) (plugging in values to get direct answer)
1:29:19 - Example 9 (limit of function with three independent variables) (plugging in values to get direct answer)
1:31:54 - Example 10 (limit of function with two independent variables) (utilizing polar identities to evaluate limit)
1:37:22 - Example 11 (limit of function with two independent variables) (using polar identities and L'Hopital's Rule to evaluate limit)
1:41:48 - Introduction to Continuity in Multivariable Calculus
1:43:53 - Definition of Continuity in Multivariable Calculus
1:49:22 - Example 12 (continuity of function with two independent variables)
1:50:26 - Example 13 (continuity of function with two independent variables) (determining domain of function)
1:51:35 - Example 14 (continuity of function with two independent variables) (determining domain of function)
1:53:33 - Example 15 (continuity of function with three independent variables) (determining domain of function) (domain includes all points except the ones that form that particular SPHERE)
1:55:47 - Introduction to composition problems doing example 16
1:58:40 - Example 17 (composition problem setting "t" equal to f(x,y)) (domain restrictions with g(t))
2:01:55 - Example 18 (composition problem) (domain restrictions with f(x,y))
2:04:59 - Introduction to Squeeze Theorem
2:06:30 - Example 19 (Squeeze Theorem)
I have been struggling with limit continuities for weeks, this video's first ten minutes cleared up everything I had wrong!
Your videos gets better and better every time. Thank You so much.
oh my god! he makes math sounds soo much fun n easy!! thank you
It's really a blessing to be taught by u. Your dedication level soars the sky.
Thank you sir for making mathematics easy for everyone one of us!
With love
From India!!
My professor for Calc 3 is named Hongwei and speaks with a hard accept from china so when he goes over anything its always a bigger challenge to not only understand the concept but him as well. So when he went over this section he sped through it only talking about the epsilon-delta definition of it and then using polar to solve some problems, never fully explaining the actual concept. This video has not only saved my grade but also my understanding of limits for all future calculations I'll be doing in my career as an engineer. Thank you so much Professor Leonard!
In my view, Professor Leonard is the Best Math Teacher online
It's funny how on calc 1, I watched your videos in regular speed. In calc 2 it was in 2x it's speed, and now I'm able to manually watch calc 3 in 3x it's speed. Just goes to show how clear it is, thanks so much for creating smooth transitions to new topics and info (:
Professor Leonard,
May god bless you with a continuous eternal function of health and wealth with no holes and asymptotes ☺😘💝
Professor Leonard, will you ever make videos for linear algebra and differential equations? I really hope you do one day!!
Procel91 he just started a differential equations series, if you still need it
Thank you much Professor Leonard for the calculus 3 videos , you such a wonderful teacher you made me like calculus , you my best teacher ever.
Your classes are helping me get through this multivariable stuff during these trying times. Thank you!
Best Calculus Professor i ever seen
As always, this is the best place on TH-cam for learning mathematics! Had some trouble at the end of the video though, and wanted to put a fact out there that tied the last section of the video up well for me: \lim_{(x,y)
ightarrow (a,b)} f(x,y) = 0 \text{ iff } \lim_{(x,y)
ightarrow (a,b)} \vert f(x,y) \vert = 0
Its LaTeX code so for anyone who cant decipher it just paste it in Overleaf or some other LaTeX editor.
A year and a half later this is still the best math lesson I've ever had
Eight years later and this is still the best math lesson I've ever had , first time watching his videos and it looks like that it won't be the last!
Dude this guy is amazing. He needs to zoom these lectures live so we are actually there. A lot of people would pay for that.
you are such an Artist....It's very satisfying when you find one of those gems on youtube!!!
Преп просто охеренный! Зачет. Cool teacher! I really enjoy his attitude. He is genuinely interested in what he is doing. On the other hand majority of university teachers barely tolerate students and see them as a nonsense and a distraction from scientific research.
I am just letting you know that I wished i found this series sooner but I found it in time for my finals. So thank you
In this first and second example if you do a 3D plot (Used Mathematica in my case) you can see that these functions are not continuous at (0,0) but has a cusp along one axis at the point (0,0) in example 1 and a cusp along the -x,y and x.y diagonals in example 2 therefore the limit is undefined. In the example lim as (x,y)->(1,0) of (2xy-2y)/(x^2 +y^2 -2x +1)Professor Leonard found that the lim as y->0 of 2y^2/2y^2 =0 presumably because this reduced to the lim y->0 of 2y^2/2y^2 is 0/0 =1 and concluded that the limit was undefined. However Mathematica finds this limit of the base limite to be 0, so the original problem has a limit of 0 also which is also what Mathematica found. This is in agreement with the plot of the surface and examining the point (x,y) = (1,0) Mathematica's determination of the original limit to be 0, so I believe his answer is differant. I guess this depends on your definition of 0/0 or it could be a bug in Mathematica. Please comment Professor Leonard and others if you can. Time point 1:11:48.
Your students are so lucky!
Thank you so much professor this means a lot for a lot of us
Your lectures are well delivered and clears out every confusion in my mind
I don't know if it's really good enough to say just a "Thank You"...but still "Thank you very much"....ur detailed conceptual explanations are a treat to watch and understand.
this guy is the best ....The way he explain Everything you can even see some logic
Professor Leonard, thank you for another well organized video/lecture on Limits and Continuity on Multivariable Functions with the classic Squeeze Theorem in Calculus Three. These concepts are introduced in Calculus One, which improve my understanding of the subject in Multivariable Calculus.
Following in the footsteps of SQ Flyer who got an A+ in math2011 intro to mul cal in 2019 fall, I have watched so many of your vids and I am going to be the next beneficiary of you!
I managed to get a past exam question correct on 1/12/2022 but my semester starts on 2/4/2022. Thanks a lot!
Im from Argentina and this is the best video i found to understand limits, ty man the only one who say that the point have to be on the path, sorry for my english.
YOU ARE AMAZING!!!!!! I HAVE CALC 2 EXAMS COMING UP IN FIVE DAYS AND YOU ARE MY SAVIOUR TO ENDING THIS COURSE WITH AN A..:)))) thanks keep this up, much better than patrick jmt
I really like your lectures. I appreciate it very much for making them available!! very learner friendly!!!
Thank you for taking the time to explain this! Amazing job! I hope that all math professors could explain the way that you do!
Give this man a medal please! Thanks for your help!
Honestly your videos help me out so much! I'd be lost without them! I can't thank you enough!!
This man is truly inspirational! loving the lectures! so helpful!
man i wish i found this channel sooner, you are like a calc legend. I might just have to move to the USA so that i could have you as my prof.
All the concepts you explained sir, you made it all look so simple. Thank you so much.
Leaned this in 2 hr when my teacher took 2 weeks and the whole class didn’t understand. Wow I am jealous of those students from your class mr calc3
this is the best professor that i've ever known :-D
you have half a million subscribers for a reason, great job!.
I'm from Pakistan . Sir I really loved your lectures and way of teaching.that's amazing 😍
Professor Leonard helped me through calc 1 with an F then round two I received a B-, then calc 2, I received a C and now Im in Calc 3 and who knows if I will pass or not! What Im telling you that just because you watch his videos and work your ass off does not mean you will automatically get an A or even pass. But don't give up people math is hard!
Goated prof, nothing made sense before this video, but even in the first 10 minutes, so many misconceptions I had were cleared up :)
i am really thankful for Professor Leonard and I even recommended your lecture for my friends and the are thankful too
reviewing again, cuz i want it to be like...hate adding... like geometry when i was young. honestly, this is where it gets exciting. also, each time i study a lesson, as i understand it even more clearly, i realize even more what a good teacher you are.
Watching from Mexico. I love your videos!
Holy shit. Literally. Those kids learning at his class must realize how blessed they are being taught by a 'true' teacher (professor). I find myself in despair watching this video at the last minute just before my midterm test. Damn. Should have known him earlier...
Thank you so much for sharing your lecture, you are truly an amazing professor!
Professor Leonard is quite simply a genius, my professor gets mad when I talk to him about this professor lol.
Excellent explanation! It's people like you who make the world go 'round
Such a wonderful professor !!!
Hey I am from india!! Your video helped me alot!!!The way u teach is very excellent!!
Thanks a mil, its amazing that you teach for two hours off of one piece of paper
Great professor, those students must be proud of him. D for done.
Thank you so much for clearing this up for me! greetings from México profesor Leonard! :)
You're a gift to humanity i just wish I could watch your lectures IRL but sadly I'm just a NTUA student here in Greece-ville. :( Cheers!
Best professor ever❤️❤️❤️
Professor Leonard, I'm always excited to see new material. I'm currently in Calc 3, and you've been a major help along the way! I have finals next week and I can understand your probably very busy yourself... Do you plan to release more videos in the next week or so? Thanks!
+Brad Nissen I'm releasing them as fast as I can make them, I hope they will help. Good luck!
+Professor Leonard when are you gonna release double and triple integral videos?
+sandile mabika
He actually already has released it, but the video was shot with a 4-D device in 4-D setting so unfortunately 3-D people are unable to see it. Maybe you should get 4-D glasses, I am selling if you are interested.
+ZnNlove Efil I'm interested.
Could you also give more examples of proving the limit DOES exist in different ways and what methods to use when?
@1:36:45 someone asked why r approaches 0 from the positive side. r is the radius of the parameterized curve, radius is a distance , so always positive and that's why approaching from the positive side ✌
this guy makes me love math. LEGEND
I am in computer engineering and this help me so much THANKS
This video is a prime example of why I never go to lectures. TH-cam can explain it 10x better in a shorter period of time. Thankfully for me, medical schools have optional lectures.
This is the best lecture on youtube! Thank you so much!
when you cant decide between watching superman or studying calculus here we are doin both
My professor didn’t e explain anything, sadly. I have an exam tonight due at 12am, and I’m here trying to understand. Now I’m stuck trying to understand everything. Thank you for helping and existing though.
Thank you so much! I'm surprised my book never mentions the squeeze theorem and instead goes through a bunch of other non-sensical stuff... You made it all perfectly clear, and the theorem kind of makes sense intuitively as well.
i think you might be wrong here. When you use for example polar coordinates and you get a number it's not strong enough to prove that the limit exists, you'd have to use that number as a candidate of a posible value of the limit and then plug it the squeeze theorem or the epsilon-delta definition.
Then just then you can prove the limit exists.
I am here because my Prof is useless and doesn't know how to actually teach... I am in love with he way Leonard teaches. So great!
You are the best, my good sir. You are the best.
Wow, first time watch your vid and learnt a lot. So clear....
Thank you for being extremely elaborate and clear!
1:01:36 Proof Professor Leonard is just a hologram.
Here too... (1:18:48)
You are amazing, Professor Leonard. Thank you.
limit DNE (3 variables) : 1:12:19
thank you best professor ever
I’m taking calculus 2 and this is so helpful, I really don’t know why the put almost all of calculus 3 in calculus 2 ...
They so don't. What? Lmao.
khaye 3endon bi amerca l calc 2 howe calc 3
He starts by talking about what we're actually doing instead of just memorizing it. It takes a little longer, but it lasts a lot longer!
The only teacher using something 3d to explain other than the white board
Thanks very much Professor Leonard
I'm thankful for your explanation sir! You make me more confident about my exams!
best professor ever