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

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

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x → 0√1+x...

Question

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

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Solution

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

Rationalising the numenator

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

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

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

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

$= \frac{2}{2} \times 0 = 0$

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

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limx→0√1+x2−√1−x2x

limx→0√1+x2−√1−x2x Rationalising the numenator limx→0(√1+x2−√1−x2)x×(√1+x2+√1−x2)(√1+x2+√1−x2) =limx→0(1+x2)−(1−x2)(√1+x2+√1−x2) =limx→02x2(√1+x2+√1−x2) =2×0(√1+√1) =22×0=0

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