Let $f_n (x) = \sin (x/n)$ for each $x \in \Bbb{R}$ and $n \in \Bbb{N}$
Calculate $f(x) = \lim _{n \rightarrow \infty} f_n(x)$ and determine the domain of $f(x)$.
I found the limit and it was $0$, but is the domain of $f(x)$ also $0?$
Also, what is the difference between $f(x)$ and $f_n(x)$?
No! Actually the range of $f$ is $\{0\}$.
$\frac{1}{n} \rightarrow 0$ and $\sin$ function is continuous to see your limit function $f(x)=0$ for all $x \in \Bbb{R}$, so the domain of $f$ is $\Bbb{R}$
For your last question ,ask yourself what is the difference between, for example, $\Big(1+\frac{1}{n}\Big)^n$ and $e$