Playing tricks with 2-variable limits

Oh hey look, a function!

0:00 Intro
1:57 Counterexamples to checking some straight lines
3:40 Counterexample to checking all straight lines
4:39 A more 'natural' counterexample (and touching grass)
6:54 Outro

The graphing program used in this video is the Grapher app that comes with Mac computers.

[Leadvest]

I'm glad I'm not the only one who finds functions in the wild like this.

[ApolloGorillaTag]

When I took Calc 3 my teacher had this as a test question: True/False If the limit exists along every straight path through a point then the limit exists at the point. (This was the final) Some unexplainable mathematical intuition told me that it had to be false I just couldn’t quite prove why, so I went with my gut and answered false. When I got my test back I had gotten everything right except for that question. I feel both a sense of intense satisfaction and anger from this video, if only I found it at the time to show the teacher!

[hyperpsych]

This is exactly what I needed to see in calc 3, I never understood how a function like that could exist but you explained it perfectly!

[paulnitro]

4:45 caught me absolutly off guard and i've started laughing out loud and every body else can`t understand why

[DarkPlaysThings]

Every upload for this channel demands time out of my day, as soon as I see the notification!

[victorscarpes]

I used to be a Calculus II tutor at university and I had a similar "trick" to test all possible line paths that didn't need to check the case x=0 separately. All I did was x=at and y=bt and took the limit with t going to zero. This method could also be generalised to allow limits that go towards any generic point (x0, y0) by having x=at+x0 and y=bt+y0 with t going to zero. Most limits would end up depending on a or b, meaning they give different results for different paths, and this, do not exist.

[JustAFriendlyFrenchDude]

The king is back ! All hail the king !

[dr.kraemer]

delightful stuff! I saw the piecewise path coming, but you surprised me with the version you found 'in the wild.' does "all curved paths" imply smoothness (i.e., C_inf)?

your board work is good, but could be leveled up by adding contrasting-color annotations to indicate transformations between lines.(also good practice in a lecture hall, but it can be hard to keep up with. time works differently here, though.) and at 6:40 you ran out of space and looped back around to the top, so by 6:47 the notation flow is ambiguous.

you showed restraint in highlighting some failure modes of graphing calculators without dunking on them. I expect you know that you could graph look-alike replicas of your functions by offsetting them a tiny bit to shift the hole in the range onto a gap in the domain lattice that's used to sample and render the surface.

[francosanson]

Absolutely immaculate humor, well done

[majorjohnson]

"Oh, a function. Oh, a function. Oh, a function..."

[NickiRusin]

At least getting all curved paths to agree is enough to prove the limit. If we work with 3-variable functions, do we need all "curved planes" to agree as well?

[musicheaven]

You know it's impressive you've made such a good video, I as a highschool student even understand it. Great job man.

[daneeko]

very nice video thats a funny function

[Wallawallawallawalla]

Is this real!? I never thought I’d see another video but I subbed last week just in case! Woo!

[timtsai]

The epsilon delta definition is exactly the point where I stopped paying attention in the calculus class

[Iearnwithme]

Loving these videos, keep it up, even if it's a little late for me to pay more attention in my Real Analysis module

[robertscholz]

Great video! There is a good problem in baby rudin that goes over this in chapter 4 I believe.

[gdmathguy]

I think for such a limit you would first try to identify how it curves (with some derivatives) and THEN try to do the limit

[RollMeAFat]

That gap at zero; would most graphing calculators just kind of, fill it in with an approximation?

[predrik]

3:00 bruh
now that I think of it I presume it's because x = 0 cannot really be represented as the form y = mx