Not necessarily but sort of. It depends on what you want to take for your starting point. Some people define ln(x)=int_1^x (1/t)dt, then you can show the properties of ln from there. Afterwards, you can then define the inverse function of ln and show it must be an exponential. You get the idea. It depends on which way you want to go, and historically logs were more familiar first and used first before exponentials because of their uses in cartography, navigation, etc. One could also argue that this limit is how one defines e to begin with. Depending on how you want to set things up again, most people are familiar being introduced to e via this limit and are given this to be the definition.
The Exponential Function e^x is realised from a binomial in two variables and has nothing to do with limit theory: ruclips.net/video/J3itUB3wGmQ/видео.html
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to prove the series of ln(1+x), you do need the derivative of ln(x) already but it's proof already requires knowing this limit is e
Not necessarily but sort of. It depends on what you want to take for your starting point. Some people define ln(x)=int_1^x (1/t)dt, then you can show the properties of ln from there. Afterwards, you can then define the inverse function of ln and show it must be an exponential. You get the idea. It depends on which way you want to go, and historically logs were more familiar first and used first before exponentials because of their uses in cartography, navigation, etc.
One could also argue that this limit is how one defines e to begin with. Depending on how you want to set things up again, most people are familiar being introduced to e via this limit and are given this to be the definition.
... Short and clear explanation ... thank you Rebecca ... best wishes, Jan-W
You don't prove you can switch lim and ln.
Besides, it's obvious that using ln you'll get an e somehow...
This is a circular proof . You can't use natural logarithm in the proof. The very defination of a natural logarithm is it's a logarithm with base e
You didn't justify that you can switch the order of ln and lim
because ln is continuous
The Exponential Function e^x is realised from a binomial in two variables and has nothing to do with limit theory:
ruclips.net/video/J3itUB3wGmQ/видео.html
Using natural logarithm makes it even funnier