A very cool limit involving e

Hi! I'm Mateo Patiño, and I record math and physics videos. Most of my content is based on problem walkthroughs and mini-lectures where I discuss a particular concept. Feel free to subscribe if you like the content or give me any feedback on things you'd like me to improve! 🍎
In this video, I will show you how to compute a limit very similar to e, but with a 2 instead of 1 in the 1/n of the original expression.
Don't hesitate to ask me any questions 😃

[@deadpoolspatan1177]

You can do it by letting N = n/2, the limit then becomes : lim (1+1/N)^2N = lim ((1+1/N)^N)^2 with N going to infinity. And this is (lim (1+1/N)^N)^2 which equals e^2 as N goes to infinity

[@btb2954]

thats what i did

[@binaryblade2]

Whole lotta math to avoid 2m = n

[@gibbogle]

"Euler" is not pronounced you-ler, it is pronounced oiler. HTH

[@miguelfelipe435]

The moment the limit becomes the derivative is worth it😃

[@archangecamilien1879]

I think I remember it was supposed to be e^2 or something, lol...that generally, the limit of (1+x/n)^n as n goes to infinity is e^x...but that's just memory...or maybe false memory, lol...

[@RSLT]

❤

[@archangecamilien1879]

Perhaps one way to do this is to use L'Hôpital's rule, lol...take the limit of the natural log of that (there must be a theorem saying the natural log of the limit is the limit of the natural log, lol)...then get rid of that n up there, take the limit using L'Hopital's rule because it's an indeterminate form, etc...infinity x 0, etc...see what that yields...

[@KingGisInDaHouse]

And the award for the cutest math teacher goes to ...this guy. I thought Flammy went uncontested for a while lol. But yeah this is how I approached the problem as well but whether or not thats actually L'Hopitals rule with extra steps is debatable.

[@Bertin-q3y]

e^2

[@gegebenein.gaussprozess7539]

Always the shame shit. It's not Yuler's number, its Oyler's. Have you ever been in a math class?

[@TheCalcSeries]

Chill buddy... Many countries outside the U.S. pronounce "Euler" as "Yuler", especially hispanic ones. I first learned about him in a math class in Spanish, so that's how I pronounce it. I believe pronunciation is a minor detail in a video focused on math, so I'd recommend you focus on that instead.

[@djttv]

How do we prove that: ln(lim f(x)) = lim ln(f(x))? It seems intuitive, but I suppose it must be proved.

[@btb2954]

Its a very obvious fact, but i dont know how to prove it 😂

[@adw1z]

If f is continuous and lim f(x) exists, then the statement follows immediately

[@adw1z]

e^x = lim n->inf (1+x/n)^n by definition, it’s when u start trying to “prove” it some other ways such as using log which is the inverse of e^x to prove e^x that u start running into circular argument problems so beware. Here you are good though, u didn’t run into any circular arguments which is impressive

[@kubakopcil9992]

Your definition is only one of the few possible definitions of e. Another useful one is that the derivative of natural logarithm is the function 1/x . This video basically proves the equivalency of the two (rather, it proves that the first one follows from the second one, but the other direction isn't too different). Just a note: Logarithm is well definite no matter what, so we can use the fact that it's inverse function to exponential.

[@RSLT]

Wow, very nice. ❤❤❤❤❤