Deriving e from the limit (1+1/x)^x as x approaches infinity

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Tambuwal Maths Class

Күн бұрын

Пікірлер: 45

@kongfung3486 26 күн бұрын

Excellent class, I can 100% digest the outcome of e, your patience and derive step by step is unbeatable.

@tambuwalmathsclass 26 күн бұрын

I so much appreciate it ❤️🙏

@Bodyknock 21 күн бұрын

1:55 A minor tweak, but once you have limit as x goes to infinity of ln(1+ 1/x) / (1/x) , you can simplify the next step a bit by substituting z = 1/x . Now you are looking at limit at z approaches 0+ of ln(1+z) / z . Using L’Hopital as in the video, the derivatives are now slightly simpler and you get just the limit at z approaches 0+ of 1 / (1 + z) which is 1. So ln y = 1, and y = e.

@Ursus-tl5lt Жыл бұрын

You cant use ln in your proof since ln is based on e and its derivative is based on the results of this theorem. The actual proof is complicated and it just proves the limit exists and is between 2 and 3, than it is calculated with taylor approximation

@ntlake Жыл бұрын

Not necessarily. You can define ln(x) without using e at all.

@blackapple62 Жыл бұрын

@@ntlake How exactly can you do this? Using the natural logarithm quite literally means you're using a logarithm of base e.

@ntlake Жыл бұрын

@@blackapple62 well, in more advanced mathematics the natural logarithm log(x) is often defined as the integral from 1 to x of dt/t. Then you prove that it's also the inverse function of exp(x).

@the.lemon.linguist 5 ай бұрын

@@ntlakeexactly you can do it by figuring it out as an area function or antiderivative of 1/x

@prakashlakhapate1598 9 ай бұрын

You can not use natural logarithm as E is to be proved. Dhanyavad

@El0melette 7 ай бұрын

you can if you define lnx as the integral from 1 to x of 1/t dt, and define e as the point where ln(x)=1.

@AlexKurian-z1s Ай бұрын

thanks man, you rock!

@isabellamichalek7705 6 ай бұрын

hello, thank you for your video, but I am confused about how you canceled the -1/x^2.

@jamesharmon4994 9 ай бұрын

What I'd be interested in knowing is what happens if 1/x is replaced by 2/x, or 3/x. Since the given formula converges and one factor is being multiplied by a constant, the new formula should also converge.

@Osirion16 2 жыл бұрын

Very cool thank you

@tambuwalmathsclass 2 жыл бұрын

Thank you

@thedeathofbirth0763 11 ай бұрын

Awesome explanation!

@DamienBlt Жыл бұрын

I understood all, but at the beguining, why there is (let y) what meaning ?, and what ln y = lim.....

@lubanlatif3713 Жыл бұрын

Basically, he assigned "y" to the equation as a variable. Therefore, we assumed that "y" equaled the whole equation. In the case of (ln y = lim), we multiplied both sides with "ln" (excluding the limit). Later, he explained it in more detail.

@DamienBlt Жыл бұрын

@@lubanlatif3713 ok thanks !

@wyboo2019 2 жыл бұрын

but isn't this limit the definition of e? or are you using the infinite polynomial definition of e^x

@tambuwalmathsclass 2 жыл бұрын

It's indeed the definition

@Grim_Reaper_from_Hell Жыл бұрын

There is only one definition lim(1+1/x)^x. It is a continuous compounding

@ЯНеДымок 6 ай бұрын

Idk, there was a n easier method imo, where by using log properties and mclaurin you not only simplify the process but also save up time.

@DB-lg5sq 4 ай бұрын

شكرا لكم على المجهودات.

@tambuwalmathsclass 4 ай бұрын

🙏🙏

@ingénieureinformatique-h5y 4 күн бұрын

Thanks

@JacobMuchebve 3 ай бұрын

🤝🤝🤝🤝🤝❤️

@AbhilashKhuntia Жыл бұрын

I am getting stuck like why are we cancelling -1/x^2 won't those two be equal to 0 and we cannot cancel 0 terms like that Please correct me if I am wrong

@nychan2939 Жыл бұрын

You can cancel equal non-zero factors from the numerator and denominator. They may be small but mustn't be zero. You can't do the cancellation if both are zero. This is the main idea in limit evaluation.

@adw1z Жыл бұрын

ln == log base e , and e is defined by this limit, so nothing was achieved other than showing consistency - there is no need to prove anything, as the definition of e itself is the limit

@ntlake Жыл бұрын

Nope. e isn't defined by this limit, it's defined by the limit as n approaches infinity of (1+1/n)ⁿ

@adw1z Жыл бұрын

@@ntlake that’s the exact same thing?? Using n,x,α,γ,.. whatever letter u want doesn’t matter as it is a dummy variable And if u mean n as in natural number or discrete limit, it doesn’t matter as the limiting function is continuous anyways, the discrete and continuous limits must agree

@ntlake Жыл бұрын

@@adw1z yeah, but no. The definition of e is the limit of a sequence, then it being equal to the limit of the function is something you have to prove. Anyway, I can define e without using this limit if you want.

@adw1z Жыл бұрын

@@ntlake the proof follows trivially directly from the definition of continuity of a function, but I’d be interested in seeing your other definition

@ntlake 11 ай бұрын

@@adw1z sorry, missed the notification. the fact that the proof is trivial doesn't change the fact that it isn't its definition. Anyway the other definition is e = 1/0! + 1/1! + 1/2! + 1/3! + ...