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
@@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).
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.
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 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.
@@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.
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
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.