How to write an epsilon-delta proof for a limit of a multivariable function

Рет қаралды 3,915

2 ай бұрын

Calculus lesson covering an example of epsilon-delta limit proof of a multivariable function.
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Пікірлер: 26

@cdkw2 2 ай бұрын

This is a stepping stone for calc 3 bprp on youtube

@sebas31415 2 ай бұрын

He said under my comment under the past vid that he wants to keep BpRp calc 1-2 (if i understood correcly but you can check the last one.)

@cdkw2 2 ай бұрын

@@sebas31415 sure

@hydropage2855 Ай бұрын

@@sebas31415GD player. And you use the same cube as me lol

@kingbeauregard 2 ай бұрын

Taking out the y^2 in the denominator, and inserting the x^2, is probably going to confuse people. The thing to remember is, one of the tricks in epsilon-delta is that you can change the expression so that it is reliably as large or larger. At that point you're doing epsilon-delta on a different expression, but because of squeeze proof logic, if that different expression satisfies epsilon-delta, so does the original. That trick is often used in conjunction with another trick: limiting delta to a narrow domain around the point in question. Sometimes the domain is absolutely arbitrary, in which case a value of 1 is common. Sometimes there is an extremum or singularity you want to avoid, in which case the size of the delta matters.

@Bedoroski 2 ай бұрын

Please make a video about 30 proofs of these multivariable functions. It'd be so valuable ❤🙏

@pedropiata648 2 ай бұрын

Agree

@pollensalta 2 ай бұрын

Beautiful explanation. Thank you very much. Please do more vids on these limits

@artursadykov3478 2 ай бұрын

Please show us in the next video the second case where the limit does not exist 😊🙏

@bprpcalculusbasics 2 ай бұрын

Oh I did that in the previous video.

@artursadykov3478 2 ай бұрын

@@bprpcalculusbasicsI mean epsilon-delta definition)

@perekman3570 2 ай бұрын

If this is calculus basics, I'd hate to see advanced topics.😂

@yassiryassir-rp4to 2 ай бұрын

why this video missing on this channel

@tobybartels8426 2 ай бұрын

You calculated this limit originally using polar coordinates, and you can get an epsilon-delta proof that follows this. From the step with 3x²|y|/(x²+y²), you note that x² ≤ x²+y² and |y| ≤ √(x²+y²), so 3x²|y|/(x²+y²) ≤ 3(x²+y²)√(x²+y²)/(x²+y²) = 3√(x²+y²) < 3δ, and finish as in the video. So where the original calculation wrote x and y exactly using r and θ, here we estimate |x| and |y| using r alone.

@wdobni 2 ай бұрын

i would like to see you solve one or two of ramanujan's more complex equations

@user-qe7sn1zp8r 2 ай бұрын

why is this unlisted?

@bitoty9357 2 ай бұрын

interesting, i didn't even know that there was an epsilon delta proof for f(x, y) funcions

@lambertwfunction 2 ай бұрын

there are epsilon delta proofs for every single metric spaces

@xinpingdonohoe3978 2 ай бұрын

Well, there are multivariable limits, and everything must be able to be done rigourously to be valid. ε-δ is simply the way we make limits rigourous.

@yoshikagekira500 2 ай бұрын

@@lambertwfunctiondo you have some videos to show how frequent are the ε-δ proofs ?

@stephenbeck7222 2 ай бұрын

@@yoshikagekira500 what do you mean by ‘how frequent’? It’s mathematical logic, you can do it with as many functions and points as you want, no limit (!).

@yoshikagekira500 2 ай бұрын

@@stephenbeck7222 in which domain of math we work with ε-δ ? Some exemples