Question : In one the proofs that I am trying to understand, the identity $$ \lim_{n \to \infty}(1 - o(1/n))^n = 1 $$ appeared and I could not understand why.
Known information : In particular, it is known $\lim_{n \to \infty} (1 - 1/n)^n = 1/e$. It is not clear how $\lim_{n \to \infty} (1 - o(1/n))^n = 1$ instead?
Background : I am new to the concept of little-o notations.
Let $f\in o(1/n)$, i.e, $nf(n)\to 0$. This also implies $f(n)\to 0$. From $\frac{\ln (1-x)}{-x}\xrightarrow[x\to 0]{}1$ it follows that $$ \lim _{n\to\infty} n\ln (1-f(n)) = \lim _{n\to\infty} -nf(n) = 0. $$ Conclude desired result from continuity of exponentiation.