Lim_x → 02x/√(a+x) √(a x) Rationalising the denominator =lim_x → 02x/(√(a+x) √(a x))×√(a+x)+√(a x)/(√(a+x) √(a x)) =lim_x → 02x(√(a+x)+√(a x))/((a+x) (a x)) =li ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x → 02 x/...

Question

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

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Solution

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

Rationalising the denominator

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

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

$= {lim}_{x \to 0} \frac{2 x \sqrt{a + x} + \sqrt{a - x}}{2 x}$

$= {lim}_{x \to 0} (\sqrt{a + x} - \sqrt{a - x}) = \sqrt{a} + \sqrt{a}$

$= 2 \sqrt{a}$

Suggest Corrections

limx→02x√a+x−√a−x

limx→02x√a+x−√a−x Rationalising the denominator =limx→02x(√a+x−√a−x)×√a+x+√a−x(√a+x−√a−x) =limx→02x(√a+x+√a−x)((a+x)−(a−x)) =limx→02x√a+x+√a−x2x =limx→0(√a+x−√a−x)=√a+√a =2√a

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