limx~0 ((x+1)^(1/x)-e)/x

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Thank you Alan,

How did you do the initial expansion?  What kind of an expansion is that?

I just used Mathcad to tell me Melody!  I'll see if I can derive it.

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Thanks Alan but worry too much, I don't understand expansions much.   :(

I only know the binomial expansion.

I just wondered :/

Here's a derivation explicitly using the binomial expansion:

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Thanks Alan,

I can follow most of that but you lost me at      $$O(x^2)$$      and what happened to all the other powers of x?

Only answer is you feel enthused because this stuff is really over my head.  I couldn't reproduce it.  :(

Rather than writing all the other terms involving x², x³ etc. I've written O(x²), meaning terms of order x² and smaller ("smaller" because we are going to make x go to 0, so x⁴, x³ etc. are going to be smaller than x²). The important thing is that all these other terms will be multiples of xⁿ, where n>1, so that when they are divided by x they are still mutiples of powers of x and hence vanish in the limit as x tends to 0.

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Thank you Alan,

I shall think on this :)