The value of x-lim2x 1-2-3x 1-3-nx 1-n2x-3 1-2-2x-3 1-3-2x-3 1-n is

Given, x→∞lim2(x )^1/2+3(x )^1/3+⋯+n(x )^1/n(2x 3 )^1/2+(2x 3 )^1/3+⋯+(2x 3 )^1/n =lim_x→∞x^1/2(2+3x^1/3x^1/2+⋯+nx^1/nx^1/2)(2x 3)^1/2(1+(2x 3)^1/3(2x 3) ...

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Byju's Answer

Standard XII

Mathematics

Sandwich Theorem

The value of ...

Question

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

\sqrt{2}

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2

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

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0

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Solution

The correct option is A $\sqrt{2}$
Given,
$lim x \to \infty \frac{2 {(x)}^{1 / 2} + 3 {(x)}^{1 / 3} + \dots + n {(x)}^{1 / n}}{{(2 x - 3)}^{1 / 2} + {(2 x - 3)}^{1 / 3} + \dots + {(2 x - 3)}^{1 / n}} = lim x \to \infty \frac{x^{1 / 2} (2 + 3 \frac{x^{1 / 3}}{x^{1 / 2}} + \dots + n \frac{x^{1 / n}}{x^{1 / 2}})}{(2 x - 3)^{1 / 2} (1 + \frac{(2 x - 3)^{1 / 3}}{(2 x - 3)^{1 / 2}} + \dots + \frac{(2 x - 3)^{1 / n}}{(2 x - 3)^{1 / 2}})} = lim x \to \infty \frac{(2 + 3 \frac{x^{1 / 3}}{x^{1 / 2}} + \dots + n \frac{x^{1 / n}}{x^{1 / 2}})}{{(2 - \frac{3}{x})}^{1 / 2} (1 + \frac{(2 x - 3)^{1 / 3}}{(2 x - 3)^{1 / 2}} + \dots + \frac{(2 x - 3)^{1 / n}}{(2 x - 3)^{1 / 2}})} = \frac{2}{\sqrt{2}} = \sqrt{2} (∵ lim x \to \infty \frac{x^{1 / 3}}{x^{1 / 2}} = 0, similarly for other terms)$

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Similar questions

Q. If

I = \int \frac{(2 x - 3)^{1 / 2}}{(2 x - 3)^{1 / 3} + 1} d x

= 3 [\frac{1}{7} (2 x - 3)^{7 / 6} - \frac{1}{5} (2 x - 3)^{5 / 6} + \frac{1}{3} (2 x - 3)^{1 / 2} - (2 x - 3)^{1 / 6} + g (x)] + C

then g(x) is equal to

Q. The value of

l i m x \to 4 \frac{(2 x)^{1 / 3} - 2}{2 x - 8}

\lim_{x \to 1} {\{\frac{x^{3} + 2 x^{2} + x + 1}{x^{2} + 2 x + 3}\}}^{\frac{1 - \cos (x - 1)}{{(x - 1)}^{2}}}

Q. The solution of

x^{1 / 3} + (2 x - 3)^{1 / 3} = [3 [x - 1]]^{1 / 3}

\sqrt{x}

\sqrt{2 x}

2 \sqrt{x}

2 \sqrt{2 x}

\dots \dots

are in geometric progression.

If the sum of first 10 terms is

31 (\sqrt{6} + \sqrt{3})

, then find the 10th term.

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Sandwich Theorem

Standard XII Mathematics

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

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