The L'H rule assumes that you know the derivative of the numerator and denominator but those derivatives are themselves obtained using first principle which uses limits .Wow!,I actually never thought of this.A good video by you highlighting that.
Using the claim that (L'Hopital's Rule is wrong) is not true. Neither is L'Hopital's Rule, in general, a redundant theorem, because it can be used for the division of functions which are antiderivatives of a known function. The title, description, and the premise of the video are all implying this. As Memelord said, the sine-cosine functions can be defined through the taylor series; sin'(x)===cos(x) follows from quite easily from the "first principles" derivative of a polynomial. L'Hopital's Rule is fine, though it is not necessary if sin(x) is defined as a power series in x. In the other differential definitions of the trigonometric functions, L'hopital's rule becomes the most direct solution. Breaking up the limit towards the end of a video makes the expression indeterminate, because the limits are infinite, and there is no single h variable any more.
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.
Yes, the quotient looks remarkably like the formula used in the definition of the derivative. That fact alone doesn't imply that it's improper to use L'Hopital's Rule. There is no circular reasoning involved. L'Hopital's Rule is presumed to have been proven for a far wider range of functions and input values before one comes to solving this problem. We don't have to reinvent the wheel ever time we want to use L'Hopital's rule for f(x)/x as x approaches zero.
"In the definition of the derivative". You just said it. That statement there already assumes understanding. I would recommend watching the whole video completely to see the key takeaway from the video. I think you completely missed the point of the video.
Another option would be to define sin(x) and cossine(x) with the complex exponencial forms, right? So sin(x) = (e^xi - e^-xi)/(2i), cos(x) = (e^xi + e^-xi)/2
@@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.
Spivack's Calculus book bypasses this issue with how they define sin and cos using areas and inverse function, so they derived the derivatives without using lim x goes to 0 of sin(x)/x
Ok im not the only one thinking like this i usually didn't tell this argument of mine to someone else because people think it's stupid but im happy to see someone is actually here nothing it
It depends on how you define sin(x) and cos(x). If you define them using Taylor series, then you can easily prove that the derivative of sin(x) with respect to x is cos(x) and so you can use l'Hopital's rule. At that point, however, it's much simpler to write sin(x)/x = (x + o(x))/x, which clearly approaches 1 as x approaches 0. AIso, in order to be able to apply l'Hopital's rule, a necessary condition is that the limit of f'(x)/g'(x) exists, which is not true in general. Take f(x) = x + sin(x) and g(x) = x as a counterexample: the limit f(x)/g(x) as x approaches infinity is an indeterminate form which can be easily shown to approach 1. However, if you try and apply l'Hopital's rule, you get the limit as x approaches infinity of f'(x)/g'(x) = 1 + cos(x), which doesn't exist.
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.
Yup, if people understand that geometric proof methodology for proving squeeze theorem first and actually understand where LH rule comes from, then yes
If you break all the problems, you ultimately end up with the axiom that you began with. Also, have you heard of godel's incompleteness theorem? It is easy to get lost in the labyrinth. Or you could just use Lhopital rule and prove it. What matters is 1. Is it a valid rule, whether we are using it properly or not. 2. Whether it is useful or not for the problem at hand. It is perfectly alright to use Lhopital rule. After all, you can find a lot of situations where you seem to go circular with but we don't have to since we already have the knowledge of it. One way or the other, you can do this with many problems. It doesn't mean that the rule is necessarily not useful, implying that it is necessarily not ok to use it despite all the conditions being satisfied.
@@NumberNinjaDave Another good example is lim x tends to 0 ((e^x -1) /x). I did watch the entire video. You explanation was correct... The only time I don't agree with you is this: I am not gonna catch people if they use the rule to do the known fundamental limits like your example or mine. Just felt like you are being overly critical. If they genuinely don't know the derivative of sinx(or e^x in my example) and got struck exactly the way you pointed out, we shall then ask them to do it with other methods like squeeze theorem, geometric method etc for better understanding. Not everyone know the fundamental results but many know the derivative. That is why I would rather let them use the rule. When I was doing 12th std(India), I got similar problems like this. It was when I was introduced to lhopital rule. Many of my friends don't even know the fundamental formula to find derivative. If you catch them, you might have to catch hold of pretty much everybody in class(on their defense, my school, just like many didn't even fully teach 11th std. I knew it at the time because I took my time and learnt the basic calculus myself. Good thing, the government intervened and stopped the sick practice right after I finished my school).
@@dhavamaneeganesh2147 That’s your perspective. On my channel, I teach understanding and not memorization. For that e limit, I would teach LH rule from the get go anyhow since now we are talking about another limit that applies the rule You’re entitled to your opinion. Thanks for chiming in 👊🥷
I'm not sure if I do it correctly To derivate sin(x) Lim h>-0 (Sin(x+h)-sinx)/h Lim h->0 (sinxcosh+sinhcosx - sinx)/h Now I consider the sine of h to be h since h Is almost 0 and sine of 0 is 0 And cosine would become 1. Lim h->0 (sinx + hcosx - sinx)/h Lim h ->0 hcosx/h So cancelling the h you get cosx
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
Who said you can split a limit of sums into a sum of limits? In this case you ignored the first and last limits, which are undefined. So I claim that step is unjustified.
@@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.
so you would have to use squueze theorem to solve the sinh/h limit? so in that case just solve the original limit using squeeze theorem? that's what i gathered.
Feel free to use L'Hopital's Rule whenever you like. It's a theorem proven in general for a large number of cases, and its proof does not depend on this specific homework problem.
@@rickdesper did you watch the whole video? It's not a matter of if you "can" use it here. It's a matter of "do you understand what you're using and where it came from before you apply it?" It's like blindly memorizing the quadratic formula without understanding where it came from. I teach understanding beyond memorization.
Quiz: LH rule is an awesome rule to simplify limits, but for the squeeze limit theorem limit from this video, why is it incorrect to use it without caution?
The L'H rule assumes that you know the derivative of the numerator and denominator but those derivatives are themselves obtained using first principle which uses limits .Wow!,I actually never thought of this.A good video by you highlighting that.
Thank you
Hence, Circular Reasoning occurs.
Using the claim that (L'Hopital's Rule is wrong) is not true. Neither is L'Hopital's Rule, in general, a redundant theorem, because it can be used for the division of functions which are antiderivatives of a known function. The title, description, and the premise of the video are all implying this.
As Memelord said, the sine-cosine functions can be defined through the taylor series; sin'(x)===cos(x) follows from quite easily from the "first principles" derivative of a polynomial. L'Hopital's Rule is fine, though it is not necessary if sin(x) is defined as a power series in x. In the other differential definitions of the trigonometric functions, L'hopital's rule becomes the most direct solution.
Breaking up the limit towards the end of a video makes the expression indeterminate, because the limits are infinite, and there is no single h variable any more.
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.
Yes, the quotient looks remarkably like the formula used in the definition of the derivative. That fact alone doesn't imply that it's improper to use L'Hopital's Rule. There is no circular reasoning involved. L'Hopital's Rule is presumed to have been proven for a far wider range of functions and input values before one comes to solving this problem.
We don't have to reinvent the wheel ever time we want to use L'Hopital's rule for f(x)/x as x approaches zero.
"In the definition of the derivative". You just said it. That statement there already assumes understanding. I would recommend watching the whole video completely to see the key takeaway from the video. I think you completely missed the point of the video.
Another option would be to define sin(x) and cossine(x) with the complex exponencial forms, right? So sin(x) = (e^xi - e^-xi)/(2i), cos(x) = (e^xi + e^-xi)/2
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.
Spivack's Calculus book bypasses this issue with how they define sin and cos using areas and inverse function, so they derived the derivatives without using lim x goes to 0 of sin(x)/x
@@ingiford175 nice
Ok im not the only one thinking like this i usually didn't tell this argument of mine to someone else because people think it's stupid but im happy to see someone is actually here nothing it
It depends on how you define sin(x) and cos(x). If you define them using Taylor series, then you can easily prove that the derivative of sin(x) with respect to x is cos(x) and so you can use l'Hopital's rule. At that point, however, it's much simpler to write sin(x)/x = (x + o(x))/x, which clearly approaches 1 as x approaches 0.
AIso, in order to be able to apply l'Hopital's rule, a necessary condition is that the limit of f'(x)/g'(x) exists, which is not true in general. Take f(x) = x + sin(x) and g(x) = x as a counterexample: the limit f(x)/g(x) as x approaches infinity is an indeterminate form which can be easily shown to approach 1. However, if you try and apply l'Hopital's rule, you get the limit as x approaches infinity of f'(x)/g'(x) = 1 + cos(x), which doesn't exist.
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.
@@NumberNinjaDave bro doesn't know what a definition is 💀
Define trig functions via the integral of area of circle or perimeter of circle and its okay to use the hospital
Yup, if people understand that geometric proof methodology for proving squeeze theorem first and actually understand where LH rule comes from, then yes
If you break all the problems, you ultimately end up with the axiom that you began with.
Also, have you heard of godel's incompleteness theorem? It is easy to get lost in the labyrinth. Or you could just use Lhopital rule and prove it. What matters is
1. Is it a valid rule, whether we are using it properly or not.
2. Whether it is useful or not for the problem at hand.
It is perfectly alright to use Lhopital rule. After all, you can find a lot of situations where you seem to go circular with but we don't have to since we already have the knowledge of it. One way or the other, you can do this with many problems. It doesn't mean that the rule is necessarily not useful, implying that it is necessarily not ok to use it despite all the conditions being satisfied.
Once we have knowledge of it…
As you just said, if you watched my entire video, you’ll notice that I say exactly just that
@@NumberNinjaDave Another good example is lim x tends to 0 ((e^x -1) /x). I did watch the entire video. You explanation was correct... The only time I don't agree with you is this: I am not gonna catch people if they use the rule to do the known fundamental limits like your example or mine. Just felt like you are being overly critical. If they genuinely don't know the derivative of sinx(or e^x in my example) and got struck exactly the way you pointed out, we shall then ask them to do it with other methods like squeeze theorem, geometric method etc for better understanding. Not everyone know the fundamental results but many know the derivative. That is why I would rather let them use the rule. When I was doing 12th std(India), I got similar problems like this. It was when I was introduced to lhopital rule. Many of my friends don't even know the fundamental formula to find derivative. If you catch them, you might have to catch hold of pretty much everybody in class(on their defense, my school, just like many didn't even fully teach 11th std. I knew it at the time because I took my time and learnt the basic calculus myself. Good thing, the government intervened and stopped the sick practice right after I finished my school).
@@dhavamaneeganesh2147 That’s your perspective. On my channel, I teach understanding and not memorization. For that e limit, I would teach LH rule from the get go anyhow since now we are talking about another limit that applies the rule
You’re entitled to your opinion. Thanks for chiming in 👊🥷
L'Hopital's rule is circular series expansion doesn't help because derivative is still needed
I'm not sure if I do it correctly
To derivate sin(x)
Lim h>-0 (Sin(x+h)-sinx)/h
Lim h->0 (sinxcosh+sinhcosx - sinx)/h
Now I consider the sine of h to be h since h Is almost 0 and sine of 0 is 0
And cosine would become 1.
Lim h->0 (sinx + hcosx - sinx)/h
Lim h ->0 hcosx/h
So cancelling the h you get cosx
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
Who said you can split a limit of sums into a sum of limits? In this case you ignored the first and last limits, which are undefined. So I claim that step is unjustified.
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.
Because cos(h)=1 for small values of h, they cancel out
so you would have to use squueze theorem to solve the sinh/h limit? so in that case just solve the original limit using squeeze theorem? that's what i gathered.
Bingo!
Feel free to use L'Hopital's Rule whenever you like. It's a theorem proven in general for a large number of cases, and its proof does not depend on this specific homework problem.
@@rickdesper did you watch the whole video? It's not a matter of if you "can" use it here. It's a matter of "do you understand what you're using and where it came from before you apply it?" It's like blindly memorizing the quadratic formula without understanding where it came from. I teach understanding beyond memorization.
@@otakurocklee did you watch the whole video? It’s wrong to use it here on this limit without understanding its origin.
Original poster, you wrote that wrong. It's sin(h)/h.
really clear , good video
Thanks so much
Always good to have some nuance
This is absolutely wrong. You should not split the terms in a limit when each of the terms which had been split is being undefined.
You need to re-review the rules of limits and how I cross-cancelled
Quiz: LH rule is an awesome rule to simplify limits, but for the squeeze limit theorem limit from this video, why is it incorrect to use it without caution?
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