Limits are...weird...for multi-variable functions | Limits along paths
ฝัง
- เผยแพร่เมื่อ 4 พ.ย. 2019
- In single variable calculus, you only had to take a limit from the left and from the right. In multi variable calculus, you can approach from every single direction. One way to say a limit doesn't exist is then to find two different paths where along those two different paths the limit approaches different values. Then for sure the limit of the function does not exist. However, you could find 10,100,1000 paths where the limit was the same along every path and yet the limit still didn't exist because you just hadn't found the one path that was weird yet. This is problematic and motivates upgrading the very definition of the limit from what we saw in first year calculus.
****************************************************
YOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!
****************************************************
►Full Multivariable Calculus Playlist: • Calculus III: Multivar...
****************************************************
Other Course Playlists:
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
***************************************************
► Want to learn math effectively? Check out my "Learning Math" Series:
• 5 Tips To Make Math Pr...
►Want some cool math? Check out my "Cool Math" Series:
• Cool Math Series
*****************************************************
►Follow me on Twitter: / treforbazett
*****************************************************
This video was created by Dr. Trefor Bazett.
BECOME A MEMBER:
►Join: / @drtrefor
MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: www.amazon.com/shop/treforbazett
How does he take something so complex and make it so simple to understand? Absolutely amazing that Dr. Bazett is able to convey so easily in a few minutes what many professors can't in hours of lecture
On god
Accidentally coming across this video just a night before my Calc3 midterm was the best thing that could happen. Cheers!
same in calc2 haha. How did your's went 2 years ago?😂
the animation really helped me understand, thank you!!!
Great "straight-forwardness" for picturing multivariable limits and great ending questions! I'm teaching multivariable calculus in the Winter and am using these as a refresher for myself. I've already seen some ways of presenting things that I'm going to use to improve my course. 👍
Thanks a lot Dr. Bazett! The graph really helped me understand intuitively.
Glad it was helpful!
@@DrTrefor U deserve alot in heaven Dr .
Great lecture with incredible effort, thanks
perhaps it would be best if some teachers could swallow their pride and instead just send a link to your videos :)
well done, i'm starting to get this now at least from a conceptual viewpoint. thank you so much!
I showed my professor (who is a great professor) his videos and told him that I use his videos to understand everything more conceptually, and he shared his channel with our entire class and linked them in our resources for the program we use.
You are excellent at explaining!
Rotational effect on the graph will be more helpful for better understanding
Thank you!! Explaining it with a actual 3-D graph helps a lot!
You're very welcome!
in which video we're gonna answer to the those questions?
thank you so much
Just found your channel.
Very nice videos and explanations!
Thank you
I used this for my midterms and I am back to it for my finals, and it is just as helpful each time (-:
Good luck!
@@DrTrefor Thank you!
I literally have your line integrals video on right now!!
no permutation or combination of 26 alphabets can define how good you teach , I'm not jokin it was actually too good
Thank you teacher. 🎉
wow !! very good graphics and explanation
Could you add the subtitles, please? You're a great teacher
Thank you so much, I understand why multivariable limit is theta independent now.
Failed out of Calculus 3 yesterday. I'll be here for the next 4 months preparing it to retake it next fall.
damn, that sucks! Welp, good luck in the fall, I'm sure you can do really well next time:)
Thank you sir 🔥
Guardian angel professor ❤
Very nice video ❤
great stuff.
Dr. Bazett, there are tricky equations with tricky points that don't have a normal vector, even if they seem smooth and well behaved. For instance, consider z=(x^3-3xy^2)/(x^2+y^2). It is easy to see that z=0 at (0, 0) , since every limit converges to that value. It is also differentiable at that point. But it has three-fold symmetry around that point, so no plane fits as tangent. How should that point be treated? In particular, normal vectors define in optics how light reflects off a mirror. If a mirror was made with that shape, how would light be reflected at that point?
loved the video 👍👍👍👍👍
this idea of restriction is just cool
Great explanation
Glad you think so!
Vamos de aprender calculo and english together, salve teacher leandro.... tudo our meu king
This video opened my eyes :OOOOOOO
Your videos are great. If you could put some translations, it would help many many more students. If you think about that, remember Portuguese, please. I want to recommend your videos to my friends, but they don't understand English. Just a suggestion.
New sub here!
This type of limit IS weird! It makes sense, though, since this particular limit is of the form 0/0. It can have multiple values and is thus indeterminate.
how can someone dislike that 😂
Sir, absyln delta definition of limit??
How does one choose which path to choose to test for continuity, anyways, maybe I choose a bunch of different paths and get the same function value for all of them (get result that limit exists for that function), but still there maybe paths where it may be different, here I do not know what I don't know.
It's a bit ambiguous
The best way maybe average contrast of each dimension will intersect at centroid then use that as objective reduce the computation in computer time because it expensive
Great content! Just a quick question, how do you prove that the limit does in fact exist? Because failing to prove that the limit does not exist does not mean that the limit exists, how would you go about proving it if plugging in values of x and y results in 0/0 or something along those lines?
i don't know answer to this question in multivariable calculus world. but in calculus 1 in university we thought to prove limits and it was only acceptable answer and just calculating wasn't acceptable. the way it work was as following:
by considering any value for instance _a_ we should show that there is a value such as _b_ where:
f(x⁰+b) < a
ultimately the job was to find a function to find _b_ based on _a_ and showing for every _a_ there is can be a _b_ founded. it was really hard to understand and solve problems in this way.
i know that using direct substitution resulting in 0/0 means you need a different approach, e.g. manipulating the function by e.g. factoring things out, L'hopital or you could try to look at the limit of the function along the line y = ax where a is just any constant in R and y = ax^2. Usually these two are enough to show that the limit does or does not exist. There is a more accurate proof for this (i think it is called the epsilon delta proof) but I'm quite unfamiliar with it. Another way is to use the intermediate value theorem and setting one variable to be greater or equal to another (this is is most commonly used when the limit of the variable goes to 0 so we can compare the two limits).
That's true about other instruments, to an extent.
what are the answers to those questions at the end?
How many of the same limit results do you need in order to conclude that the limit exists. For example taking the 2 paths allong the x axis and the y axis is not enough right?
Even a million paths wouldn't be enough. That's why we have to either use some proven algebraic rules or prove it from the limit definitions directly. It's not easy!
we learned this topic today , the way we chose was to use squezee theorem
Sir, why we can't prove it by y=mx, if limit is free from m. Please sir make video on this topic. I am going to die for this question.sir please, please,
Sir, is there a condition that the curve along which we approach should be smooth? Because otherwise there can be many path along which any limit would not exist
so you can make y = any type of 'x' ??
Please postprocess audio to remove this empty room echo effect. Beside this your content is great :)
What software are you using for the 3D graphs?
Matlab
let's consider this function f(x,y)=x*y/((x-1)^2+y^2) and (x,y) approach to (1,0) then what path can we take? Can we still take y = x or should we take for example y= x -1?
wow it's math pewdiepie. damn he explains stuff so well...
I wondered about Question 2; my intuition says that, yes, a limit would exist if the approach on every straight line yields the same result.
My thought was that if you approach the limit with all combinations of straight line paths and they all converge to the same value then approaching along a curved path would also yield the same value and the limit would exist. May be there is more to it though.
so the limits kinda like a volume/ area
That is green theory was introduced but I called it là grange theory
Sir, how can we contact with you.
What is the difference between the limit and the continuity definition? One could say in the continuity definition we add the last term L(left)=L(right)="F(xo)"and in the limit definition we dont use the last term thats the only thing and thats wrong thats why if we are asked about this exact example about continuity at the origin the question has no sense at all. And what about if we are asked about the continuity in general of
f=xy/(x^2+y^2) of course it is!!!
tbh, i came to find answers to the questions myself.
Anyways, thanks.
Can someone plz explain why the 0,0 of y and x arent a same point ?
Another time unbelievable content
However at 2min54 if you plug y=-x the denominator will not be wero ?
Sir, please make a video on last two questions. Sir please😭😭😢😥😭😭😭.
Sir, if limit is same with all paths along the line, would it be same along any curvy path.
1:15
Trefor, I think path can't be curvature that is not continuous (because that would cause problems for h --> 0), but I am not sure. Also, if curvature is differentiable, then (for h --> 0) we can aproximate it by some streight line and that we can easly solve. So, does curvature (path) must be continuous and/or differentiable? Thanks
who are these 5 people? :)
I just love your videos. Is there a way to take you out on a date ? 💓
4:15
multivariable function... such a hypocrite!!!
He’s hot I can’t focus
XnotYas i cant stop staring at his beard