First slide

Introduction to limits

Question

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

Moderate

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

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