what happens if you make the substitution n = 1/x?
That's not the case, it is definitely the limit to infinity: lim n to infinity (1+1/n)^n - Wolfram|Alpha
If you want choose a suitable large value of and compare it to e.
By first inspection you notice you get the indeterminate form therefore you can apply l'Hopital's rule, that means:
Calculating gives:
Calculating gives:
So now you have this limit:
which is equal to (because the limit of a product is the product of the limits of the separated factors):
And
(recheck that!)
Therefore the only thing we have to do is to calculate:
You notice you get again the indeterminate form therefore we can apply l'Hopital's rule again:
Therefore the limit is