Lim x - 0-1-x-x2-x-1-2 x2

Lim_x → 0√(1+x+x^2) √(x+1)/2x^2 Rationalising the numerator =lim_x → 0(√(1+x+x^2) √(x+1))× (√(1+x+x^2)+√(x+1))/2x^2 (√(1+x+x^2)+√(x+1)) =lim_x → 0(1+x+x^2) (x+1 ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x → 0√1+x...

Question

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

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Solution

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

Rationalising the numerator

$= {lim}_{x \to 0} \frac{(\sqrt{1 + x + x^{2}} - \sqrt{x + 1}) \times (\sqrt{1 + x + x^{2}} + \sqrt{x + 1})}{2 x^{2} (\sqrt{1 + x + x^{2}} + \sqrt{x + 1})}$

$= {lim}_{x \to 0} \frac{(1 + x + x^{2}) - (x + 1)}{(\sqrt{1 + x + x^{2}} + \sqrt{x + 1})}$

$= {lim}_{x \to 0} \frac{x^{2}}{2 x^{2} (\sqrt{1 + x + x^{2}} + \sqrt{x + 1})}$

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

$= \frac{1}{2 \times 2} = \frac{1}{4}$

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limx→0√1+x+x2−√x+12x2

limx→0√1+x+x2−√x+12x2 Rationalising the numerator =limx→0(√1+x+x2−√x+1)×(√1+x+x2+√x+1)2x2(√1+x+x2+√x+1) =limx→0(1+x+x2)−(x+1)(√1+x+x2+√x+1) =limx→0x22x2(√1+x+x2+√x+1) =12(√1+√1) =12×2=14

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