The value of limx→∞ 2x1/2+3x1/3+4x1/4+…nx1/n(2x−3)1/2+(2x−3)1/3+…+(2x−3)1/n is

A
0
B
2
C
$\sqrt{2}$
D
$\frac{1}{\sqrt{3}}$

Solution:

Let

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

Then,

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

$\Rightarrow l = lim_{h \to 0} \frac{2 + 3 h^{(\frac{1}{2} - \frac{1}{3})} + 4 h^{(\frac{1}{2} - \frac{1}{4})} + \dots + n h^{(\begin{array}{ll} 1 1 \\ 2 n \end{array})}}{(2 - 3 h)^{2} + (2 - 3 h)^{\frac{1}{3}} h^{(\frac{1}{2} - \frac{1}{3})} + \dots + (2 - 3 h)^{n} h^{(\frac{1}{2} - \frac{1}{n})}}$

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

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

Solution:

Related content