Question

Evaluate: $\underset{n\to \infty }{lim}[\{1+{\left(\frac{1}{n}\right)}^4\}{\{1+{\left(\frac{2}{n}\right)}^4\}}^{1/2}{\{1+{\left(\frac{3}{n}\right)}^4\}}^{1/3}{\{1+{\left(\frac{4}{n}\right)}^4\}}^{1/4}....2^{1/n}]$

Answer 1

$\underset{n\to \infty }{lim}[\{1+{\left(\frac{1}{n}\right)}^4\}{\{1+{\left(\frac{2}{n}\right)}^4\}}^{1/2}.....{\{1+{\left(\frac{n}{n}\right)}^4\}}^{1/n}]$

as $n\to \infty ;\frac{1}{n}\to 0$

$=\{1+0\}{\{1+0\}}^{1/2}.....{\left\{2\right\}}^0$

$=1\times 1\times 1\times ...\times 1=1$