There's a problem with that solution 'cause binomial theorem works with natural numbers, hence, when computing the limit, you assume that is natural since is a real number.
First of all, why don't you directly factor by in the beginning instead of factorizing with exponentials ?
Also, what if ?
Plus, you ought to know the limits and , which are proved using the definitions of derivative numbers (if not "advanced" calculus...)
You factorize later, after changing into exponentials, while it would have been exactly the same if you did it from the beginning.To use known limits.
when ?Why should I consider that case?
What's the interest of asking to prove a simple formula if it has to use nested limits ?By just knowing that those limits are easy to prove.
Well, I'm no one to tell you this, so just ignore it if you want.
Krazalid is correct to say limit definition is no good if is not an integer.
And generalized binomial theorem is faulty since you implicity use the power rule in its derivation.
The standard way is to notice that by definition for .
Here is the inverse function for .
If we are familar with the basic rules for and just apply the chain rule and get your answer.