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

Given $\underset{x\to \infty }{\lim }\frac{2x^{1/2}+3x^{1/3}+\mathrm{....}+n{x}^{1/n}}{{(2x-3)}^{1/2}+{(2x-3)}^{1/3}+\mathrm{....}+{(2x-3)}^{1/n}}$$\underset{h\to 0}{\lim }\frac{2\left(\frac{1}{h^{1/2}}\right)+3\left(\frac{1}{h^{1/3}}\right)+\mathrm{....}+n\left(\frac{1}{h^{1/n}}\right)}{\frac{1}{h^{1/2}}{(2-3h)}^{1/2}+\frac{1}{h^{1/3}}{(2-3h)}^{1/3}+\mathrm{.....}+\frac{1}{h^{1/n}}{(2-3h)}^{1/n}}$ $\left[onputtingx=\frac{1}{h}asx\to \infty ,h\to 0\right]$ $=\underset{h\to 0}{\lim }\frac{2+3h^{\left(\frac{1}{2}\frac{1}{3}\right)}+4h^{\left(\frac{1}{2}\frac{1}{4}\right)}+\mathrm{.....}n{h}^{\left(\frac{1}{2}\frac{1}{n}\right)}}{{(2-3h)}^{1/2}+h^{\left(\frac{1}{2}\frac{1}{3}\right)}{(2-3h)}^{1/3}+\mathrm{.....}+h^{\left(\frac{1}{2}\frac{1}{n}\right)}{(2-3h)}^{1/n}}$ $=\frac{2+0+0+0+\mathrm{.....}}{2^{1/2}+0+0+0+\mathrm{.......}}=\sqrt{2}$