_ x (x^2+x+1-x^2+1 )= - Vidyadip

MCQ

$\lim _{x \rightarrow \infty}\left(\sqrt{x^2+x+1}-\sqrt{x^2+1}\right)=$

A
$-\frac{1}{2}$
✓
$\frac{1}{2}$
C
1
D
-1

✓

Answer

Correct option: B.

$\frac{1}{2}$

(B)
$\lim _{x \rightarrow \infty}\left(\sqrt{x^2+x+1}-\sqrt{x^2+1}\right)$
$=\lim _{x \rightarrow \infty} \frac{x^2+x+1-x^2-1}{\sqrt{x^2+x+1}+\sqrt{x^2+1}}$
$=\lim _{x \rightarrow \infty} \frac{1}{\sqrt{1+\frac{1}{x}+\frac{1}{x^2}}+\sqrt{1+\frac{1}{x^2}}}$
$=\frac{1}{1+1}=\frac{1}{2}$

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