Limx- 01-x-x 2-x-12 - 2

Limx #8594;01+x+x2 x+12x2 It is of the form 00. Rationalising the numerator: nbsp;limx #8594;01+x+x2 x+11+x+x2+x+11+x+x2+x+12x2 = nbsp;limx #8594; ...

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Byju's Answer

Standard XII

Mathematics

Integration by Partial Fractions

limx→ 01+x+x ...

Question

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

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Solution

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

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

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

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

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

= $\frac{1}{(\sqrt{1 + 0 + 0} + \sqrt{0 + 1}) \times 2}$

= $\frac{1}{2} \times \frac{1}{2}$

= $\frac{1}{4}$

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limx→0 1+x+x2-x+12x2

limx→01+x+x2-x+12x2 It is of the form 00. Rationalising the numerator: limx→01+x+x2-x+11+x+x2+x+11+x+x2+x+12x2 = limx→01+x+x2-x+11+x+x2+x+12x2 = limx→0x21+x+x2+x+12x2 = 11+0+0+0+1×2 = 12×12 = 14

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