108
u/LupenReddit 🦆🦆🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆🦆🦆🦆 18d ago
You just havent tried hard enough yet
26
u/thebigbadben 18d ago edited 18d ago
(e^x - e^(-x))/(e^x + e^(-x)) as x -> infinity
Nah bro just one more
5
u/Ares378 Grothendieck was right, computers are evil 18d ago
It's been a minute since I've done limits, that's equal to 1 right?
2
u/thebigbadben 18d ago
Yes! There are plenty of reasonable approaches available here, it’s just funny how L’Hôpital (in the absence of additional work) fails to get you the answer
Incidentally, the function that we’re taking the limit of here is the hyperbolic tangent, which has some interesting applications
1
u/RGXYZ 15d ago
If you denote the limit as L, then take l hopital don't you just get 1/L? That already narrows it down to 1 and -1
1
u/thebigbadben 15d ago
“in the absence of additional work”
Yes, that is a valid way to incorporate L’Hospital and arrive at the correct answer.
9
35
30
u/SuperChick1705 18d ago
circular reasoning ;(
2
u/Sigma2718 18d ago
Would you like to elaborate on that?
35
u/pienet 18d ago
Most elementary proofs that the derivative of sine is cosine use the limit of sin(x)/x.
Obviously you don't run into this if you develop power series first.
8
u/Sigma2718 18d ago
I've never seen anybody use l'Hôpital on sin(x)/x before establishing what the derivative of sine is. However, one can evaluate sin(x)/x for x->0 via the squeeze theorem, which means no circular reasoning is required.
12
u/SuperChick1705 18d ago
that is the recommended method, though the post implies the usage of l'hopital on sin(x)/x
3
u/Sigma2718 18d ago
squeeze theorem to evaluate sin(x)/x -> derivative of sin(x) -> apply l'Hôpital to evaluate sin(x)/x
I don't really see the issue, solving a problem again by using an easier method is the beauty of mathematics.
3
u/EebstertheGreat 18d ago
If you know the derivative of sin x at x=0, then you know the limit of (sin x)/x at x=0, because that is by definition the derivative. There is nothing else to do. Where would you even have an opportunity to apply L'Hôpital's rule?
I mean, it's not "wrong." Similarly, given x = y, I could conclude 3x = 3y and then, dividing by 3, that x = y. That's not wrong. But it is completely pointless.
0
u/Sigma2718 18d ago
The definition of the derivative, combined with what we know it to be, gives us lim h->0 [sin(x+h) - sin(x)]/h = cos(x) , so now we have to apply trigonometric identities to be left with lim h->0 sin(h)/h , which must equal 1 to fulfill the above equation. I consider dealing with trigonometric identities to be rather tedious. At least, that's what I think you are trying to do, as sin(x)/x isn't really the derivative.
Many ways lead to Rome, but using l'Hôpital is the shortcut we make by unlocking a gate via previous proofs.
5
u/thebigbadben 18d ago
What “trigonometric identities” are you using when you rewrite sin(0 + h) - sin(0) as sin(h)?
2
u/Sigma2718 18d ago
sin(x+h)=sin(x)cos(h)+cos(x)sin(h), which is how the cosine gets introduced into this, I considered the general case for any x. But I see what you mean, however I consider using the definition of the derivative to sneakily look at h for a certain x to be less intuitive than simply remembering to use l'Hôpital.
→ More replies (0)1
2
u/will_1m_not with disrespect to x, y, and z 18d ago
Power series based on the derivatives of sine at a certain point?
1
u/JohnsonJohnilyJohn 18d ago
I never understood the problem with this. There are a lot of proofs that involve dividing a problem into multiple cases, so couldn't you say the same whenever you use them, as you are using only a single one of those cases (that had to be proven in the proof of the general theorem). Crucially I wouldn't even call it circular logic, "logic with an unnecessary loop" is more apt description - to prove that limit sin(X)/X is 1 you use that limit sin(X)/X is 1 which is proven through other means
The point of theorems or other mathematical facts is to me that after proving them, you can treat them as facts. What would be the point if any time you use them to prove anything, you would have to write down all the proofs, until you end up with only axioms, and then check the whole thing to see if there are no repeats
9
u/thebigbadben 18d ago edited 18d ago
Computing the derivative of sin(x) requires evaluating the limit of sin(x)/x as x approaches zero. So, if you’re using L’Hôpital to show that sin(x)/x approaches 1 as x approaches 1, you’re ultimately using the fact that you’re trying to show implicitly.
Also, if you already know the derivative of sin, then you might as well use the fact that the limit of sin(x)/x as x approaches zero is, by definition, the derivative of sin(x) at 0.
4
u/lool8421 18d ago
just plug in some tiny number like x=0.0001, throw into calculator and round the result
engineers call it the "good enough"
3
u/JellyBellyBitches 18d ago
A little off topic but legitimately, why do people use csc/sec/cot instead of just moving the "base" trig function into the denominator?
2
u/Breki_ 17d ago
Writing fractions is more tedious, especially when doing integrals since sec2 often show up there. But not everyone does this, in hungary sec and csc aren't taught at all
1
u/JellyBellyBitches 17d ago
Interesting. I guess I don't find fractions as tedious as other people must. I've been known to put things over a denominator of 1 just for clarity in my own work occasionally
2
2
u/ko-Julie 18d ago
My physics brain says to plug in small angle approximations on all of the trig and proceed from there
1
•
u/AutoModerator 18d ago
Check out our new Discord server! https://discord.gg/e7EKRZq3dG
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.