Answer
$\sqrt[4] e$
Work Step by Step
We are given the sequence:
$\left\{\sqrt{\left(1+\dfrac{1}{2n}\right)^n}\right\}$
Rewrite the general term of the sequence:
$\sqrt{\left(1+\dfrac{1}{2n}\right)^n}=\sqrt{\left(\left(1+\dfrac{1}{2n}\right)^{2n}\right)^{1/2}}=\left(\left(1+\dfrac{1}{2n}\right)^{2n}\right)^{1/4}$
Use the fact that $\lim\limits_{n \to \infty} \left(1+\dfrac{1}{n}\right)^n=e$ to determine the limit of the given sequence:
$\lim\limits_{n \to \infty} \left(\left(1+\dfrac{1}{2n}\right)^{2n}\right)^{1/4}=\left(\lim\limits_{n \to \infty} \left(1+\dfrac{1}{2n}\right)^{2n}\right)^{1/4}=e^{1/4}=\sqrt[4] e$