Lim x - 1-3-x-5-x-x2-1

Lim_x → 1√(3+x) √(5 x)/x^2 1 [It is of (0/0) form] Rationalising the numerator =lim_x → 1(√(3+x) √(5 x))/(x 1)(x+1)×(√(3+x)+√(5 x))/(√(3+x)+√(5 x)) =lim_x → 1{3 ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x → 1√3+x...

Question

${lim}_{x \to 1} \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^{2} - 1}$

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Solution

${lim}_{x \to 1} \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^{2} - 1}$ [It is of $(\frac{0}{0})$ form]

Rationalising the numerator

$= {lim}_{x \to 1} \frac{(\sqrt{3 + x} - \sqrt{5 - x})}{(x - 1) (x + 1)} \times \frac{(\sqrt{3 + x} + \sqrt{5 - x})}{(\sqrt{3 + x} + \sqrt{5 - x})}$

$= {lim}_{x \to 1} \frac{{3 + x - (5 - x)}}{(x - 1) (x + 1) (\sqrt{3 + x} + \sqrt{5 - x})}$

$= {lim}_{x \to 1} \frac{- 2 + 2 x}{(x - 1) (x + 1) (\sqrt{3 + x} + \sqrt{5 - x})}$

$= {lim}_{x \to 1} \frac{- 2 (x - 1)}{(x - 1) (x + 1) (\sqrt{3 + x} + \sqrt{5 - x})} (\frac{0}{0})$

$= {lim}_{x \to 1} \frac{- 2}{(1 + 1) (\sqrt{3 + 1} + \sqrt{5 - x})}$

$= \frac{2}{(1 + 1) (\sqrt{3 + 1} + \sqrt{5 - x})}$

$= \frac{2}{(2) (2 + 2)} = \frac{1}{4}$

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limx→1√3+x−√5−xx2−1

${lim}_{x \to 1} \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^{2} - 1}$