Consider the function f(x)= (1+1/x)^x . My aim is to prove that this function is bounded and its limit exists as x approaches infinity, for all positive real x. Now, I am aware of many hand waving proofs available online that simply state that the limit of f(x) is equal to e: Such proofs merely invoke the use of L Hopital's rule and are circular in nature. Unfortunately, I have not been able to find a completely rigorous proof online(is it because the level of difficulty is too high??), so I thought it would be appropriate for me to ask my question here.
What I have managed to prove so far:
1) f(x) is monotone increasing over the domain (0,inf)
2) If we merely consider the sequence x(n)=(1+1/n)^n then I can find a upper bound of 3 and then show, by the Monotone Convergence Theorem, that the sequence converges to a unique limit, which, we define to be e.(Correct me if I am wrong, I might have left out some details here).
3) However, in the case of the continuous function f(x), I am unable to prove that its limit at infinity exists, I don't know how to find a upper bound for f(x). If I could, I am done, since I can just apply MCT. Or, alternatively, I can just show that the supremum of f(x)=e, from which we are done immediately without having to invoke MCT. So, I was just wondering: Is it possible to provide a completely rjgorigo proof of the fact that f(x) has an upper bound, or perhaps even the limit at infinity of f(x) exists? I may be a little confused here due to my relatively insufficient knowledge, so any clarification will be much appreciated.
Attached below is a scheme of thought from my teacher, who is unable to prove the last two points. I don't know if the train of thought is logical though?
What I have managed to prove so far:
1) f(x) is monotone increasing over the domain (0,inf)
2) If we merely consider the sequence x(n)=(1+1/n)^n then I can find a upper bound of 3 and then show, by the Monotone Convergence Theorem, that the sequence converges to a unique limit, which, we define to be e.(Correct me if I am wrong, I might have left out some details here).
3) However, in the case of the continuous function f(x), I am unable to prove that its limit at infinity exists, I don't know how to find a upper bound for f(x). If I could, I am done, since I can just apply MCT. Or, alternatively, I can just show that the supremum of f(x)=e, from which we are done immediately without having to invoke MCT. So, I was just wondering: Is it possible to provide a completely rjgorigo proof of the fact that f(x) has an upper bound, or perhaps even the limit at infinity of f(x) exists? I may be a little confused here due to my relatively insufficient knowledge, so any clarification will be much appreciated.
Attached below is a scheme of thought from my teacher, who is unable to prove the last two points. I don't know if the train of thought is logical though?
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