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Q)

$\lim\limits_{x\to \infty}[x-\sqrt{x^2+x}]$=

$(a)\;\large\frac{1}{2}$$\qquad(b)\;1\qquad(c)\;\large\frac{-1}{2}$$\qquad(d)\;0$

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A)
Given limit =$\lim\limits_{x\to \infty}\large\frac{x^2-(x^2+x)}{x+\sqrt{x^2+x}}$
$\Rightarrow \lim\limits_{x\to \infty}\large\frac{-x}{x(1+\sqrt{1+1/x)}}$
$\Rightarrow \lim\limits_{x\to \infty}\large\frac{-1}{1+\sqrt{1+1/x}}$
$\Rightarrow \large\frac{-1}{2}$
Hence (c) is the correct answer.
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