The value of x-lim - -x-x-x-x -is-

=x→∞lim [ √(x+√(x+√(x))) √(x) ] =x→∞limx+√(x+√(x)) x/√(x+√(x+√(x))+√(x))=x→∞lim√(x+√(x))/√(x+√(x+√(x))+√(x)) =x→∞lim√(x) ( 1+1/√(x) )^1/2/√(x) [ ( 1+1/√ ...

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Byju's Answer

Standard XI

Mathematics

Indeterminate Forms

The value of ...

Question

$The value of l i m x \to \infty [\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}] is .$

1/2

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-1/2

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Solution

The correct option is A
1/2

$= l i m x \to \infty [\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}] = l i m x \to \infty \frac{x + \sqrt{x + \sqrt{x}} - x}{\sqrt{x + \sqrt{x + \sqrt{x}} + \sqrt{x}}} = l i m x \to \infty \frac{\sqrt{x + \sqrt{x}}}{\sqrt{x + \sqrt{x + \sqrt{x}} + \sqrt{x}}} = l i m x \to \infty \frac{\sqrt{x} {(1 + \frac{1}{\sqrt{x}})}^{1 / 2}}{\sqrt{x} ⎡ ⎢ ⎣ {(1 + \frac{1}{\sqrt{x}} \sqrt{1 + \frac{1}{\sqrt{x}}})}^{1 / 2} + 1 ⎤ ⎥ ⎦} = \frac{1}{1 + 1} = \frac{1}{2}$

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The value of limx→∞[√x+√x+√x−√x]is.

The correct option is A 1/2 =limx→∞[√x+√x+√x−√x] =limx→∞x+√x+√x−x√x+√x+√x+√x=limx→∞√x+√x√x+√x+√x+√x =limx→∞√x(1+1√x)1/2√x⎡⎢⎣(1+1√x√1+1√x)1/2+1⎤⎥⎦=11+1=12

$The value of l i m x \to \infty [\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}] is .$