The value of $\underset{x\to \mathrm{\infty }}{lim} \frac{2x^{1/2}+3x^{1/3}+4x^{1/4}+\dots n{x}^{1/n}}{(2x-3{)}^{1/2}+(2x-3{)}^{1/3}+\dots +(2x-3{)}^{1/n}}$ is

Solution:

Let

$l=\underset{x\to \mathrm{\infty }}{lim} \frac{2x^{1/2}+3x^{1/3}+4x^{1/4}+\dots +n{x}^{1/n}}{(2x-3{)}^{1/2}+(2x-3{)}^{1/3}+\dots +(2x-3{)}^{1/n}}$

Then, 

$l=\underset{h\to 0}{lim} \frac{2h^{-1/2}+3h^{-1/3}+4h^{-1/4}+\dots +n{h}^{-1/n}}{(2-3h{)}^{1/2}{h}^{-1/2}+(2-3h{)}^{1/3}{h}^{-1/3}+\dots +(2-3h{)}^{1/n}{h}^{-1/n}}$, where $h=\frac{1}{x}.$

$\Rightarrow l=\underset{h\to 0}{lim} \frac{2+3h^{\left(\frac{1}{2}-\frac{1}{3}\right)}+4h^{\left(\frac{1}{2}-\frac{1}{4}\right)}+\dots +n{h}^{\left(\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}n\end{array}\right)}}{(2-3h{)}^2+(2-3h{)}^{\frac{1}{3}}{h}^{\left(\frac{1}{2}-\frac{1}{3}\right)}+\dots +(2-3h{)}^n{h}^{\left(\frac{1}{2}-\frac{1}{n}\right)}}$

$\Rightarrow l=\frac{2}{\sqrt{2}}=\sqrt{2}$