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

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

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

@@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).

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

Thank you

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.

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

@@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

​@@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.

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

@@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! + ...

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.

@@lubanlatif3713 ok thanks !

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.

It's indeed the definition

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