the COOLEST limit on YouTube!
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- เผยแพร่เมื่อ 27 ก.ย. 2024
- We will evaluate the limit of ln(x)/W(x) as x goes to infinity. This will add one more nice connection between the natural log function and the production function. To see a crash course on the Lambert W function, check out: • Lambert W Function (do...
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I substituted x with ue^u so I will get (u+ln(u)) /u which its limit goes to 1
That’s an extremely smart way to do it!!!
that's what I did too !
It genuinely threw me off that he didn't do that. We have a W(x), we probably will make life easier by getting rid of it.
Same method I got! (I suspect a ton of people did it this way as well...)
Everyone here who did not do this obvious but BRILLIANT approach should be ashamed! Edit: _I'm_ ashamed!
Where is thr fish?
The fish: 🐟
my reply got deleted lol, google free speech platform.
google deleted both my fish lol, frxe-spexch = orangeman? lol.
It swam away
GREEN PENCIL???
I was shocked also lol
Yup, legends says that when BPRP uses the FOURTH mighty color will be a signal of The Advent
Yooo we need more "FISH" vids. (the w function)
bprp should change his name to bprpgp 😂
At least include the blue pen first!!
it should be bprpbpgp, not just bprpgp! Unless the b at the beginning stands for both black and blue at the same time
if x+y=8, find the max of x^y (Lambert W function)
th-cam.com/video/zdAJXil-NvA/w-d-xo.html
We got green pen on bprp before GTA6 release
Imagine how fun GTA6 will be after SweatBabbyInk "fixes" it! lolfml.
ohhhh google canceled facts again lol.
Nah his old vids had green pen
Now I just need to see THE PURPLE PEN!
Wait until he pulls out the orange pen
Where's your *"PurplePen"* from the old videos? XD
Ahh yes! Welcome to another very cool video of *"BlackpenRedpenBluepenGreenpen"* litterelly
I was curious and checked inverses of x^n*exp(x) and apparently all of them also behave like ln(x)
the natural next question: limit of (ln(x) - W(x)) / ln(ln(x)) as x -> infinity
It's 4am why am I watching this lol. Notification gang?
Are you at US?
@Naman_shukla410 probably central US, maybe Mexico or Central America. Most likely US though.
It was 10am here when he posted the video.
9:18 missaying: he want to say 1/W(x) goes to zero as z goes to inf.
oh yeah baby show me the limit
@9:45: I too think this is really, really cool. There can never be too much Lambert W function content on TH-cam. Now that you've computed the derivative of W(x), can you compute its antiderivative? I'll give it a try, and leave another comment if I succeed.
It took me a couple of hours, but I finally got it. On my first attempt, I solved W'(x) = W(x)/(x*(W(x)+1)) for W(x) to get x*W'(x)/(1-x*W'(x)), and integrated x*W'(x)/(1 - x*W'(x))*dx through a series of substitutions, starting with u = x*W'(x) (which introduced an exponential in W(x) to remove an x), culminating in a polynomial in t, times e^t. Unfortunately, I must have made a sign error somewhere, because the result did not have a derivative equal to W(x) (instead getting (W(x)^2 + 2*W(x) - 1)/(W(x) + 1) - 1). But it was close enough that I was able to deduce that the true antiderivative of W(x) was likely a quadratic in W(x), times e^W(x), plus a constant, and starting from there, I was able to find h(x) = (W(x)^2 - W(x) + 1)*e^W(x) + C, which is a function such that h'(x) = W(x).
Of course, after a few more minutes of playing around with this, I realized I should have *started* with the substitution u = W(x), because that would give me dx = (u+1)*e^u*du, and integrating u*(u+1)*e^u*du is easy....
Se armó la grande en TH-cam.
The 🐟 Function is here!
Can you show please how to compare W(W(1)) and (W(1))^2 without calculator?
Nothing better than a fresh limit on a Saturday morning
Yes!!!
Hi
Lets go, comeback of the lambert W function
While the limit is correct, these functions do not really become the same at large x. For large x, W(x)=ln(x)-ln(ln(x)+O(1). Hence as x->Infinity, W(x)/ln(x) ->1 because ln(x) grows faster than ln(ln(x)). However, as x-> infinity also ln(x)-W(x) -> ln(ln(x)) -> Infinity. Thus the difference between log and product log becomes infinite at large x. It's just that this difference grows slower than the functions themselves, so the result of dividing them tends to 1 at large x...
Fun challenge: what's the minimum of W(x)/ln(x)? Yes, it has a "nice" answer.
@@ingobojak5666 e/(e+1)?
My thought before substituting is to just let x -> xe^x. Then we have lim x->inf (lnx + x)/x = 1
Limits can never be cool!
This is really really cool.
Woag
I love the LambertW. It holds a special place with me, since highschool, leading me on a wonderful goose chase.
Does this work in general with inverses of functions like this?
If f(x) goes to infinity as x goes to infinity and g(x)=xf(x), will their inverses always have this limit?
09:33 - This plot with (x, y) confused me, then I made similar plot with (exp(x), y), and now it's obvious.
Before viewing, I guessed e^(1/e), which is actually not that far off! :)
Whenever i see the w function i automatically i lose interest. I am not sure what you fixation with it is. Its not something that's thought here and we are lucky to be spare of it.
This is the first time I got an idea of a real world property of the W fuction. Thanks!
i thought he was making a rap video for a moment when he kept saying "to the e to the y"
It is not a "natural log 🪵"! It is a "natural logarythm".
Black pen red pen blue pen green pen YAY
7:58 surprise, he have a green pen
64000 Rutherford Curve
9:16 Vai me dar zero? Não é infinito?
who is that in the photo kept behind?
nice
How gosh he got a green one ! 😮
You’re the goat BPRP
Neat!
x>inf (de hospital)(1/x)/W(x)/x(W(x)+1)=(W(x)+1)/W(x)..>1
I tried and done in 2nd try ❤
Why is this so good?
So good!
very coooool
Painis
❤❤
So good to know this, because the Lambert W() function has been mysterious to me.
What about a limit or an integral with logarithms in variable base? For example logx(some function in x)
You can simplify log_x(f(x)) to ln(f(x)) / ln(x).
Why is the domain [-1,inf)? xe^x accepts any number as input. Maybe i just dont kniw what "to have inverse" means exactly
A function f has an inverse if for every x there is a unique y so that f(y) = x. For that to happen, it has to be bijective - one-to-one and onto.
The function x e^x can be defined for all real x, but you’ll find that there are values of x less than -1 and values greater than -1 that give the same value of the function, meaning you can’t pick a unique inverse across that domain. By restricting the domain of the function to [-1,infinity), you force it so that there’s only one value in the domain that corresponds to each value in its range.
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Now can you compute this: lim ( ln(x)-W(x) )
x→∞
ingobojak5666 already answered that in his comment.
No that is not true. The +1 with the infinity makes it a limit question again. Those sums do eventually diverge and if you use very large numbers to look at them like a Graham's number then the natural log wins with the greater growth.
?
@@blackpenredpen well we are looking not only at the limit originally, but a limit of limits. while infinity and and infinity+1 are both infinity they are not equal. You had another infinity over infinity and you needed to perform L'H again. We cannot draw a conclusion when it is infinity over infinity. That +1 will matter as if you look that the delta between the changes of change the ln while exceedingly the product log is even slower. As you like to say you have to do more work.