The value of limx→∞2(x)1/2+3(x)1/3+4(x)1/4+.....+n(x)1/n(2x−3)1/2+(2x−3)1/3+.....+(2x−3)1/n is

Given limx→∞2x1/2+3x1/3+....+nx1/n(2x−3)1/2+(2x−3)1/3+....+(2x−3)1/nlimh→02(1h1/2)+3(1h1/3)+....+n(1h1/n)1h1/2(2−3h)1/2+1h1/3(2−3h)1/3+.....+1h1/n(2−3h)1/n [on putting x=1h as x →∞, h→0] =limh→02+3h(1213)+4h(1214)+.....nh(121n)(2−3h)1/2+h(1213)(2−3h)1/3+.....+h(121n)(2−3h)1/n =2+0+0+0+.....21/2+0+0+0+.......=2

The value of limx→∞2(x)1/2+3(x)1/3+4(x)1/4+.....+n(x)1/n(2x−3)1/2+(2x−3)1/3+.....+(2x−3)1/n is

Given limx→∞2x1/2+3x1/3+....+nx1/n(2x−3)1/2+(2x−3)1/3+....+(2x−3)1/nlimh→02(1h1/2)+3(1h1/3)+....+n(1h1/n)1h1/2(2−3h)1/2+1h1/3(2−3h)1/3+.....+1h1/n(2−3h)1/n [on putting x=1h as x →∞, h→0] =limh→02+3h(1213)+4h(1214)+.....nh(121n)(2−3h)1/2+h(1213)(2−3h)1/3+.....+h(121n)(2−3h)1/n =2+0+0+0+.....21/2+0+0+0+.......=2

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

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$\frac{1}{\sqrt{3}}$

$\sqrt{2}$

answer is .

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Detailed Solution

Given $\lim_{x \to \infty} \frac{2 x^{1 / 2} + 3 x^{1 / 3} + .... + n x^{1 / n}}{{(2 x - 3)}^{1 / 2} + {(2 x - 3)}^{1 / 3} + .... + {(2 x - 3)}^{1 / n}}$ $\lim_{h \to 0} \frac{2 (\frac{1}{h^{1 / 2}}) + 3 (\frac{1}{h^{1 / 3}}) + .... + n (\frac{1}{h^{1 / n}})}{\frac{1}{h^{1 / 2}} {(2 - 3 h)}^{1 / 2} + \frac{1}{h^{1 / 3}} {(2 - 3 h)}^{1 / 3} + ..... + \frac{1}{h^{1 / n}} {(2 - 3 h)}^{1 / n}}$ $[o n p u t t i n g x = \frac{1}{h} a s x \to \infty, h \to 0]$ $= \lim_{h \to 0} \frac{2 + 3 h^{(\frac{1}{2} \frac{1}{3})} + 4 h^{(\frac{1}{2} \frac{1}{4})} + ..... n h^{(\frac{1}{2} \frac{1}{n})}}{{(2 - 3 h)}^{1 / 2} + h^{(\frac{1}{2} \frac{1}{3})} {(2 - 3 h)}^{1 / 3} + ..... + h^{(\frac{1}{2} \frac{1}{n})} {(2 - 3 h)}^{1 / n}}$ $= \frac{2 + 0 + 0 + 0 + .....}{2^{1 / 2} + 0 + 0 + 0 + .......} = \sqrt{2}$