The value of _ x → ∞ 2 (x) ^ 1 / 2 + 3 (x) ^ 1 / 3 + 4 (x) ^ 1 / 4...

Given,

$\lim_{x \to \infty} \dfrac{2{x}^{1 / 2} + 3{x}^{1 / 3} + . . . . + n{x}^{1 / n}}{{\left(2x - 3\right)}^{1 / 2} + {\left(2x - 3\right)}^{1 / 3} + . . . + {\left(2x - 3\right)}^{1 / n}}$

$= \lim_{h \to 0} \dfrac{2\left(\dfrac{1}{{h}^{1 / 2}}\right) + 3\left(\dfrac{1}{{h}^{1 / 3}}\right) + . . . . + n\left(\dfrac{1}{{h}^{1 / n}}\right)}{\dfrac{1}{{h}^{1 / 2}}{\left(2 - 3h\right)}^{1 / 2} + \dfrac{1}{{h}^{1 / 3}}{\left(2 - 3h\right)}^{1 / 3} + . . . . . + \dfrac{1}{{h}^{1 / n}}{\left(2 - 3h\right)}^{1 / n}}$

\left[Onputtingx = \dfrac{1}{h}asx \to \infty, h \to 0\right]$<br /><br />$= \lim_{h \to 0} \dfrac{2 + 3{h}^{\left(\dfrac{1}{2} - \dfrac{1}{3}\right)} + 4{h}^{\left(\dfrac{1}{2} - \dfrac{1}{4}\right)} + . . . . + n{h}^{\left(\dfrac{1}{2} - \dfrac{1}{n}\right)}}{{\left(2 - 3h\right)}^{1 / 2} + {h}^{\left(\dfrac{1}{2} - \dfrac{1}{3}\right)}{\left(2 - 3h\right)}^{1 / 3} + . . . . + {h}^{\left(\dfrac{1}{2} - \dfrac{1}{n}\right)}{\left(2 - 3h\right)}^{1 / n}}$<br /><br />$= \dfrac{2 + 0 + 0 + 0 + . . . . . . .}{{2}^{1 / 2} + 0 + 0 + . . . . . . .} = \sqrt{2} .