Mathematicians on Torn?

0/0 is undefined or a so-called indeterminate.

It makes however sense to speak of a limit of a quotient where the numerator and denominator both tend towards zero. In these cases, one applies certain theorems or other tricks to find the limit.

I'm studying BEng Mathematics with Computer Science at University, worst decision of my life xD

I think he is probably referring to limits. In that case refer to L'Hospital.

Yes, that's why I told him that one applies certain theorems.

Also, does a high school student know L'Hopital? It's overkill to use L'Hopital on that level. He's better off using factorization or similar tricks to evaluate limits.

Well on a high school level L'Hospital couldn't be easier...
Most of the functions you work with can be derivated (sorry, English isn't my native language so I'm not sure if that is actually the correct way to say it) incredibly easy, so I don't see a reason why it shouldn't be taught.

And sorry, I completely missed the part where you talked about the limits. XD
Not too good with reading thoroughly it seems.

My native language isn't English too. So, no worries :)

Sure, most of the functions in high schools are differentiable at most points. But let's say the high school student asked for the limit of sinx/x when x turns towards zero. That gives a 0/0 indeterminate. Would you ask a high school student to use L'Hopital on that?

In my opinion, students in high school have to pretty careful applying L'Hopital because of similar problems. Therefore, tricks of the trade like factorization, squeeze lemmas etc are much more appropriate until the student thoroughly learns calculus.

Maybe I overlooked something there, but I honestly don't see the issue. As far as I am aware you can just simply apply L'Hospital and then either think a minute about the problem and get the solution.

sin'(x) = cos(x) and x' = 1. At this point the function would read cos(x)/1. For x -> 0, cos(x) -> 1. So lim (x -> 0) sinx/x = 1.

And while there are certainly more complicated ways of solving this I still think that a High School Student would have enough knowledge to be taught this method. There are a lot of examples where it makes sense to use L'Hospital and you cannot go wrong if you already know it.

...

If you use L'Hopital to compute sinx/x when x->0, how do you prove that sin'(x)=cos(x)?

The prove is actually pretty simple. But even without being able to prove it, a high school student should now the derivatives of sin(x) or cos(x). I mean its a basic circle.

sin'(x) = cos(x)
cos'(x) = -sin(x)
-sin'(x) = -cos(x)
-cos'(x) = sin(x)

As to the prove: Once you have the correct approach its quite simple.

f(x) = sin(x)

f'(x) = lim(y -> 0) [sin(x+y) - sin(x)/y]

Then all you do is rewrite sin(x+y) using the trigonometric addition theorem and then you should be able to get to f'(x) = cos(x) by just simplifying the formula as much as possible. (If you want all the steps you let me know and ill do them in excel and paste a screenshot on here. Using just text to write out formulas isn't too good as they wont be so easy to rad.

Right. And when you write out sin(x+y)-sin(x) you get sinx*cosy+siny*cosx-sinx. Then you get:

(lim y->0)(sinx*(cosy-1)/y+cosx*(siny/y))

Now, you have to compute the limit of siny/y when y->0. If you use L'Hopital on this limit, then you would be using a circular argument. So, you have a problem. You are trying to prove that (sinx)'=cosx, so you cannot assume this (as we would with L'Hopital). This gives us a necessity to not apply L'Hopital and be able to evaluate limits without it. (Here you can use a squeezing argument).

If you have any questions, feel free to ask.

I am not sure if you simply mistyped by accident (like I said, functions are very confusing when written out in one line without much formatting ability) but as far as I can tell it should be [...] + (cosx * siny)/y.

[image: i.imgur.com]

Oh, I get what you are trying to say now. You mean I cannot prove to a high school student that sin'(x) = cos (x) because I cannot use L'Hospital to prove this?

If that is the case I really don't see how that makes sense... Firstly, the original argument is still, could a high school student apply L'Hospital to a function like sin(x)/x for lim x-> 0. So to that argument the obvious answer is yes. Any high school student knows the derivatives of sin and cos, regardless of the fact if he can prove it or not.

As to the circular argument: There are multiple ways to prove that the derivatives of sin. The way I explained is just the only one I would expect a High School Student to be able to follow.

I doubt anyone at high school level would be worried about proofs about derivatives of trig functions. When I was in High School in Germany, I learned L'Hospital rule without proof, the same way that I learned that the derivatives of sin(x) is cos(x).
Only when you get to University, you start to prove all these things rigorously.

Hence for these purposes, I don't think it's too much to talk about L'Hospital and derivatives, as it is probably the easiest way to find these limits, as it can be applied to a wide range of situations and you don't need any tricks. (and again, that's what I have been taught in high school as well)

I disagree. Proofs are not something separated from mathematics. They are an integral part of it. The classic proof of sinx/x when x-->0 should be seen by every high school students. Of course, you can say that in University you'd learn why something is true, but high school mathematics is not supposed to only be able to use algorithms. By not learning why L'Hopital is true og why derivative of sine is cosine you're not doing mathematics, you're following orders.

It seems that high school math differs. If German students learn mathematics in high school without proofs, I feel sorry for them. At least I learned it in high school.

I didn't mistype. There is no difference between cosx*(siny/y) and (cosx*siny)/y.
In general, a*(b/c)=(ab)/c.

f**k me what did I learn in high school.....

Like sin(x)/x as x tends to 0?

Edit: I see people have already brought this up.

https://www.youtube.com/watch?v=v8rZOl6jeeo

Hold up here, maybe I'm just not familiar with the name but I've never heard of L'Hopital... and I'm a 4th year Mechanical Engineer...

Looking at it briefly it looks like you're discussing calculus with x->0 (First principles), is this correct?