Limx #8594;02 x 2+xx It is of the form 00. Rationalising the numerator: nbsp;limx #8594;02 x 2+x2 x+2+xx2 x+2+x = nbsp;limx #8594;02 x 2+xx2 x+2+x ...

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Standard XI

Mathematics

Sequence

lim x → 0 2-x...

Question

$\lim_{x \to 0} \frac{\sqrt{2 - x} - \sqrt{2 + x}}{x}$

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Solution

$\lim_{x \to 0} [\frac{\sqrt{2 - x} - \sqrt{2 + x}}{x}]$

It is of the form $\frac{0}{0}$ .
Rationalising the numerator:

$\lim_{x \to 0} [\frac{(\sqrt{2 - x} - \sqrt{2 + x}) (\sqrt{2 - x} + \sqrt{2 + x})}{x (\sqrt{2 - x} + \sqrt{2 + x})}]$

= $\lim_{x \to 0} [\frac{(2 - x) - (2 + x)}{x (\sqrt{2 - x} + \sqrt{2 + x})}]$

= $\lim_{x \to 0} [\frac{- 2 x}{x (\sqrt{2 - x} + \sqrt{2 + x})}]$

= $\frac{- 2}{2 \sqrt{2}}$

= $- \frac{1}{\sqrt{2}}$

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limx→0 2-x-2+xx

limx→02-x-2+xx It is of the form 00. Rationalising the numerator: limx→02-x-2+x2-x+2+xx2-x+2+x = limx→02-x-2+xx2-x+2+x = limx→0-2xx2-x+2+x = -222 = -12

$\lim_{x \to 0} \frac{\sqrt{2 - x} - \sqrt{2 + x}}{x}$