Відео

Hahahahaha

NGL I thought this was gonna be a 1 min prank vid and that he was gonna do exactly this 😄

@@numbers93 lol

Oh that’s epic! I’ve never used this in a statistical application

@@NumberNinjaDave well when i had to i had no choice but to use it, and there we go

Very true, 🥷

💪💪💪💪

🥷

But how does that solve the original problem of an indeterminate form?

@@NumberNinjaDave If you use this definition (which is somewhat problematic, since it defines a real operation with complex numbers), the derivative of sine and cossine follow from the derivative of the exponencial. In this case you can use L'Hôpital to calculate this limit, because it isn't circular reasoning anymore.

Another way of thinking about it, is that the “c” is just a function that has a derivative of 0 along all the places where f(x) is defined. But it can still suddenly change values where f(x) and F(x) is undefined.

That’s a piecewise definition that is essentially how the domain of the absolute value in ln | x| + C is defined

Very true!

Thanks so much

Yup, if people understand that geometric proof methodology for proving squeeze theorem first and actually understand where LH rule comes from, then yes

Mistake on my part!

Thank you for the catch! Good eye 🥷

Very close. It’s missing a small but important detail

@@NumberNinjaDave ha ha , you mean const. C ( C is going to Chill )

@@youngman3544 Yes sir 🥷

Feel free to go ahead and do that to show us your work

@@NumberNinjaDave d/dx(x^x) = x(x^(x-1)) = x(x^x)(1/x) = x^x d/dx(x^x)= x^x(lnx) 1st+2nd ans= x^x+x^xlnx

@@kiddo0142 this method happens to get the answer but isn’t a sound approach First 1/x needs to be taken with caution as what if x is 0? Second, the first half didn’t really make progress. It just landed back at the original problem

@@NumberNinjaDave well if x=0 the original problem would be undefined as 0^0 and this is just a trick if you need to impress someone :)

@@kiddo0142 0^0 is defined as 1 (some schools of thought say undefined, but I’m from the school of thought that there’s only one element in the power set of the empty set and so it’s an agreed on multiplicative identity). But your method allows division by 0 in the 1/x step.

Heck yeah 🥷

@@NumberNinjaDave Just subbed

@@nonchalantguy9461 i appreciate you 🥷

Absolutely, ninja! That works (and I use that technique too)

Glad you enjoyed it, 🥷

Bingo!

You are indeed a ninja 🥷 your approach is how I would actually do it too if I saw this on an exam

Taylor series shows us just that and squeeze theorem, that when centered around 0 and for very small x, sin x is roughly x. But I don’t use Taylor series here since that too uses the definition of the derivative already

@@NumberNinjaDave so... Is it the correct way or not?

@@cucler6718 your math looks correct and gets the right answer…question for you though. Do you think that approach is the right approach?

@@NumberNinjaDave I really don't think it's the right approach lol

@@cucler6718 agreed

Yes, I think it was stated that in the upper half of the domain, you can drop the absolute sign because radians are above the X-Axis, making the result positive, so an absolute value sign wouldn’t be needed in the first half of the integral.

@@nonchalantguy9461 correct, but not for the left half

@@NumberNinjaDave Ahhh. Makes sense

All valid points! However, keep in mind that Taylor series stem from the same condition of knowing the nth derivative and the main takeaway of this video as mentioned in the last few seconds was to understand the derivation and usage of LH rule before just blindly using it. Students should know how to solve this squeeze theorem limit without LH rule. Otherwise, it’s just blind formula memorization and I teach understanding, not memorization.

@@NumberNinjaDave I didn't say to use Taylor series, I said to *define* sin(x) with its "Taylor series". That is, define sin(x) = x - x²/2 + x³/6 - x⁴/24 + ... This means that you don't need derivatives anymore to get to that power series, since you made it true by definition. Many papers prefer to adopt the power series definition or the integral definition for some functions, since it often simplifies calculations.

@@ntlake and again, that definition comes from derivatives. You are solving for the constants needed to satisfy an infinite term Taylor series. Go look up where the specific Taylor series definition of the sin function comes from.

@@NumberNinjaDave it's not a Taylor series anymore, it's a power series definition, even though it's often called "Taylor series definition" because that's where the idea came from. It makes no sense at all to say "that definition comes from derivatives", a definition doesn't come from anything. It's a definition. Defining sin(x), cos(x) and many other functions through power series is common practice in modern calculus. One can then show that the power series definition is consistent with their Taylor series, but that's another thing.

Thank you! Yeah, just personal preference

You can definitely split them. Yeah, I definitely ignored them since the middle one was sufficient on its own to prove the point of the video

@@berryesseen infinity isn’t a number so we aren’t changing the limit. It’s unbounded. You certainly can.

@@berryesseen if you say so 😉 you can disagree all you want. Looking at your channel, it looks like you’re bashing my channel in an attempt to draw viewers to your channel and to troll. That tells me everything I need to know.

The Taylor series build off first principle indeed. But again, their prerequisite stems from already having used the first principle to begin with The title isn’t to say that always using LH rule is wrong. It’s to say it’s wrong with the provided problem. The provided problem is the squeeze limit problem that was used to define LH Rule to begin with, so naively using LH rule in this case is simply wrong. As for solving the problem provided, the Taylor series approach does show us one possible solution of 1 since when centering around 0, sin x converges to approximately x when x is super small Of course, the other way to solve and prove is geometrically but not what this video is about.

​@@NumberNinjaDaveI have to disagree. If you define derivatives differently (e.g. using Grassman numbers), then L'Hôpital's rule does not depend on any limits

@@Erreniumyou can disagree all you want. It’s circular reasoning to use LH rule on the same squeeze limit theorem problem that was used to define the rule in the first place. No need to define anything special to change the fact

@@NumberNinjaDave "It’s circular reasoning to use LH rule on the same squeeze limit theorem problem that was used to define the rule in the first place." That's their point tho, you can just define it in such a way that doesn't depend on the squeeze limit theorem. There are ways to define derivatives without limits (granted they are probably way past the level of your standard Calculus I course...), from which you can get the derivative of sin(x) without using the limit you're trying to solve

@@NumberNinjaDave It's not circular reasoning, where's the circle. It's just reasoning that doubles back on itself. "lim x->0 sin(x)/x = 1 by the squeeze theorem, therefore d/dx sin(x) = cos(x), therefore lim x->0 sin(x)/x = 1 by LH" is somewhat redundant (we already knew that limit from earlier) but entirely valid. It would be *circular* if you tried to prove it as "well d/dx sin(x) = cos(x), because lim x->0 sin(x)/x = 1 by LH, because d/dx sin(x) = cos(x), because lim x->0 sin(x)/x = 1 by LH, because [...]", because the argument keeps going forever, and proofs have to be finite. If going "downwards" through the proof terminates at the squeeze theorem, then it's finite and therefore valid.

Watch my latest video published today and you’ll see why I didn’t use LH rule here.

@@NumberNinjaDave OK, I think you should have warned that we couldn't use that "magic" tool, the explanation has no sense.

@@DanielMartinez-ss5co thanks for the feedback, man 🙏

Thank you

Toucan Sam 🦅

It was cross-grouping. It wasn’t so much flipping that single fraction as it was grouping terms almost like in an “x” across fractions

Lol

@@NumberNinjaDave I hope you can laugh more, here: Place an infinite amount of points on a circumference of a circle. Then pick any point of your choice on the circumference. Add one to that point or subtract one from that point. How far have you moved on the circumference in radians?

@@user-ky5dy5hl4d sounds like you already know the answer 😊

@@NumberNinjaDave I love physics. But unfortunately one cannot do anything on physics without mathematics. But I am not a mathematician by any means. I just dabble in math. My answer is such that if you add one on that circumeference you will not move at all because infinity is not a number. But one more question (from physics). What causes the speed of light?

@@NumberNinjaDave Sir! I thank you for your input. I am learning from you. I had had heavy calculus years ago. But now people like you retreive my mind about math. I believe that you are an ingredient to have an element about time which can be expressed in math. You are great!

Glad to help!

No. When given a problem like this, we have to demonstrate that either it exists (and what it is), or prove it doesn’t. When you see a question like this on your AP exam, you may not know

Yup, that works! For very small x super close to 0, sin x approximates to x (also as seen from the Taylor series)

Exactly. If you use LHR here, you’re essentially using the definition of LHR here from squeeze theorem on itself so it’s more precise to not use LHR

@@tolkienfan1972 it’s incorrect to use the rule here because it’s circular logic using the definition on itself. Research it. I’m aware of LHR and you can watch my other videos on it but it’s incorrect to use it here

@@tolkienfan1972 a candidate, yes, as it’s in indeterminate form, and for other function types to solve, indeed. But the presented limit here already is using a squeeze theorem method to solve the limit of the sin trigonometric function By definition, LHR was derived using the squeeze theorem to observe the convergence of sin x / x for very small x There really isn’t anything more to debate on this. Just like chat gpt hallucinations, we can’t simply quickly read something on Wikipedia or WA and run away with it as a confirmation bias. We need to look at the full problem at hand, and if you researched even further, you’ll see math forums discussing this QED

@@tolkienfan1972 no it isn’t sufficient in this specific case of squeezing a trig function which is where lhr came from in the first place

Best comment of the day. Terry Silver welcomes you

Reverse chain rule and u sub are analogous.

What if a student forgets the formula on an exam? I teach understanding and not memorization

Yes, a non constant and how to solve it

Very nice 👍

1

@@manav_chak7410 well done, ninja 🥷