$\displaystyle \begin{align*} \lim_{n \to \infty} \left[ \left( 2n \right) ^{\frac{1}{n}} \right] &= \lim_{n \to \infty} \left\{ e^{ \ln{ \left[ \left( 2n \right) ^{\frac{1}{n}} \right] } } \right\} \\ &= \lim_{n \to \infty} \left[ e^{\frac{1}{n} \ln{(2n)} } \right] \\ &= \lim_{n \to \infty} \left[ e^{ \frac{\ln{(2n)}}{n} } \right] \\ &= e^{ \lim_{n \to \infty} \left[ \frac{\ln{(2n)}}{n} \right] } \\ &= e^{\lim_{n \to \infty} \left( \frac{\frac{1}{n}}{1} \right) } \textrm{ by L'Hospital's Rule} \\ &= e^{ \lim_{n \to \infty} \left( \frac{1}{n} \right) } \\ &= e^0 \\ &= 1 \end{align*}$