Answer
converges to $x$
Work Step by Step
As we know that when $x \gt 0$, $\lim\limits_{n \to \infty} (1+\dfrac{x}{n})^{n}=e^x$
Let $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} (\dfrac{x^n}{2n+1})^{1/n}$
This implies that $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} (\dfrac{x^n}{2n+1})^{1/n}$
and $a_n= (x) \lim\limits_{n \to \infty} (\dfrac{1}{2n+1})^{1/n}=(x) \lim\limits_{n \to \infty} e^{1/n} \ln (\dfrac{1}{2n+1})$
and $(x)(e^{0})=x$
Thus, $\lim\limits_{n \to \infty} a_n=x$ and {$a_n$} converges to $x$.
