Q101 / 160 · AP EAMCET 2022 (07 Jul Shift 1) · Mathematics

$\lim _{x \rightarrow 0} \frac{\sqrt{11+|x|-6 \sqrt{2+|x|}}}{6-2 \sqrt{2+|x|}}=$

  • A
    $-1$
  • B
    $-\frac{1}{2}$
  • C
    $\frac{\sqrt{11-6 \sqrt{2}}}{3-\sqrt{2}}$
  • Correct answer
    D
    $\frac{1}{2}$

Solution

$\lim _{x \rightarrow 0} \frac{\sqrt{11+|x|-6 \sqrt{2+|x|}}}{6-2 \sqrt{2+|x|}}$ $=\lim _{x \rightarrow 0} \frac{\sqrt{3^2+(\sqrt{2+|x|})^2-6 \sqrt{2+|x|}}}{6-2 \sqrt{2+|x|}}$ $\begin{aligned} & =\lim _{x \rightarrow 0} \frac{\sqrt{(3-\sqrt{2+|x|})^2}}{2(3-\sqrt{2+|x|})} \\ & =\lim _{x \rightarrow 0} \frac{(3-\sqrt{2+|x|}}{2(3-\sqrt{2+|x|})}=\frac{1}{2}\end{aligned}$

Hence, the correct answer is option: 'D'.

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