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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits2 Marks
MCQ
$\lim _{x \rightarrow 0} \frac{\sqrt{2+x}-\sqrt{2-x}}{x}=$
  • A
    $\frac{1}{2 \sqrt{2}}$
  • ✓
    $\frac{1}{\sqrt{2}}$
  • C
    2
  • D
    $\sqrt{2}$
✓

Answer

Correct option: B.
$\frac{1}{\sqrt{2}}$
(B)
$\lim _{x \rightarrow 0} \frac{\sqrt{2+x}-\sqrt{2-x}}{x}$
$=\lim _{x \rightarrow 0} \frac{(\sqrt{2+x}-\sqrt{2-x})(\sqrt{2+x}+\sqrt{2-x})}{x(\sqrt{2+x}+\sqrt{2-x})}$
$=\lim _{x \rightarrow 0} \frac{2}{\sqrt{2+x}+\sqrt{2-x}}$
$=\frac{2}{\sqrt{2+0}+\sqrt{2-0}}=\frac{1}{\sqrt{2}}$
Alternate Method:
$\lim _{x \rightarrow 0} \frac{\sqrt{a+x^n}-\sqrt{a-x^n}}{x^n}=\frac{1}{\sqrt{a}}$
$\lim _{x \rightarrow 0} \frac{\sqrt{2+x}-\sqrt{2-x}}{x}=\frac{1}{\sqrt{2}}$

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