convergence of altered sequence

Question

Let $x_n=(1+\frac{y}{n})^n$. It is well known that $x_n\rightarrow e^y$ as $n\rightarrow \infty$. Now, if we consider $X_n=\left(1+\frac{y}{n}+o(\frac{1}{n})\right)^n$, then it makes sense that $X_n\rightarrow e^y$, too, because $o(1/n)$ vanishes faster than $1/n$ as $n\rightarrow\infty.$ However, how can this be shown rigorously? Does this also hold more generally? I.e., if $f(1/n)\rightarrow z$, do we then also have $f(1/n+o(1/n))\rightarrow z$?

Answer 1

Well, $X_n=\left(1+\frac{y}{n}+o(\frac{1}{n})\right)^n=\left(1+\frac{y}{n}\right)^n\cdot\left(1+o(\frac{1}{n})\right)^n$, so its limit may be treated as a product of two limits which are both (supposedly) easy to find.