Limx #8594;01+x 1 x2x It is of the form 00. Rationalising the numerator: nbsp;limx #8594;01+x 1 x1+x+1 x2x1+x+1 x = nbsp;limx #8594;01+ #160;x ...

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Quadratic Formula

lim x → 01+x-...

Question

$\lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{2 x}$

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Solution

$\lim_{x \to 0} [\frac{\sqrt{1 + x} - \sqrt{1 - x}}{2 x}]$

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

Rationalising the numerator:

$\lim_{x \to 0} [\frac{(\sqrt{1 + x} - \sqrt{1 - x}) (\sqrt{1 + x} + \sqrt{1 - x})}{2 x (\sqrt{1 + x} + \sqrt{1 - x})}]$

= $\lim_{x \to 0} [\frac{(1 + x) - (1 - x)}{2 x (\sqrt{1 + x} + \sqrt{1 - x})}]$

= $\lim_{x \to 0} [\frac{2 x}{2 x (\sqrt{1 + x} + \sqrt{1 - x})}]$

= $\frac{1}{\sqrt{1 + 0} + \sqrt{1 - 0}}$
= $\frac{1}{2}$

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