Lim x - 0-1-3 x-1-3 x-x

Lim_x → 0√(1+3x) √(1 3x)/x Rationalising the numerator =lim_x → 0(√(1+3x) √(1 3x))/x×(√(1+3x)+√(1 3x))/(√(1+3x)+√(1 3x)) =lim_x → 0(1+3x) (1 3x)/(√(1+3x)+√(1 3x ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x → 0√1+3...

Question

${lim}_{x \to 0} \frac{\sqrt{1 + 3 x} - \sqrt{1 - 3 x}}{x}$

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Solution

${lim}_{x \to 0} \frac{\sqrt{1 + 3 x} - \sqrt{1 - 3 x}}{x}$

Rationalising the numerator

$= {lim}_{x \to 0} \frac{(\sqrt{1 + 3 x} - \sqrt{1 - 3 x})}{x} \times \frac{(\sqrt{1 + 3 x} + \sqrt{1 - 3 x})}{(\sqrt{1 + 3 x} + \sqrt{1 - 3 x})}$

$= {lim}_{x \to 0} \frac{(1 + 3 x) - (1 - 3 x)}{(\sqrt{1 + 3 x} + \sqrt{1 - 3 x})}$

$= {lim}_{x \to 0} \frac{6 x}{(\sqrt{1 + 3 x} + \sqrt{1 - 3 x}) x} (\frac{0}{0})$

$= {lim}_{x \to 0} \frac{6}{(\sqrt{1 + 3 x} + \sqrt{1 - 3 x})}$

$= \frac{6}{\sqrt{1} + \sqrt{1}}$

$= \frac{6}{2} = 3$

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Similar questions

Evaluate $l i m x \to 0 \frac{(\sqrt{1 + 3 x} - \sqrt{1 - 3 x})}{x}$

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lim x \to 0 \frac{\sqrt{1 + 3 x} - \sqrt{1 - 3 x}}{x}

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lim x \to 0 \frac{\sqrt{1 + x^{2}} - \sqrt{1 - x + x^{2}}}{3^{x} - 1}

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lim x \to 0 {(\frac{1^{x} + 2^{x} + 4^{x}}{3})}^{\frac{1}{x}}

lim x \to 0 {(\frac{2^{x} + 3^{x} + 8^{x} + 27^{x}}{4})}^{\frac{1}{x}}

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lim x \to 0 {⎡ ⎣ \frac{(\frac{1}{2})^{x} + (\frac{1}{3})^{x} + (\frac{1}{8})^{x} + (\frac{1}{27})^{x}}{4} ⎤ ⎦}^{\frac{1}{x}}

lim x \to \infty {[\frac{3 x - 4}{3 x + 2}]}^{\frac{x + 1}{3}}

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limx→0√1+3x−√1−3xx

limx→0√1+3x−√1−3xx Rationalising the numerator =limx→0(√1+3x−√1−3x)x×(√1+3x+√1−3x)(√1+3x+√1−3x) =limx→0(1+3x)−(1−3x)(√1+3x+√1−3x) =limx→06x(√1+3x+√1−3x)x(00) =limx→06(√1+3x+√1−3x) =6√1+√1 =62=3

${lim}_{x \to 0} \frac{\sqrt{1 + 3 x} - \sqrt{1 - 3 x}}{x}$