Euler's Number to determine congruence and limits Establish the convergence and find the limits of $(1+ (\frac{1}{1+n} ))^ n$ and $(1-(\frac{1}{n} ))^ n$.
I understand that these can be written as $\frac{(1+ (\frac{1}{1+n} ))^{n+1}}{(1+ (\frac{1}{1+n} ))}$ and $\frac{(1+(\frac{1}{n-1} ))^{-1(n-1)}}{(1+ (\frac{1}{n-1} ))}$. I also know that the limits are $e$ and $ \frac {1}{e}$ respectively as the denominators' limits go to $1$ as $n \to \infty$.
My question is how is it that $(1+ (\frac{1}{1+n} ))^{n+1}$ and  $(1+(\frac{1}{n-1} ))^{n-1}$ both have limits $e$ based on the original derivation of $e$ from $(1+ (\frac{1}{n} ))^ n$.

Given that you know $$lim_{n\to \infty} (1+ (\frac{1}{n} ))^ n = e$$ and since we know that the $$lim_{n\to \infty}f(n) = lim_{n\to \infty}f(n+1)$$ then $$lim_{n\to \infty} (1+ (\frac{1}{1+n} ))^{n+1} = e$$, and similarly for the other one. 
