Lim_x→∞{√(x+1) √(x)}√(x+2) =lim_x→∞ [ √(x+1) √(x) ] [ √(x+1)+√(x) ]/ [ √(x+1)+√(x) ]×√(x+2)×√(x+2)/√(x+2) =lim_x→∞(x+1 x)/√(x+1)+√(x)×(x+2)/√(x+2) =lim_x→∞(x+2) ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x →∞√x+1-...

Question

${lim}_{x \to \infty} {\sqrt{x + 1} - \sqrt{x}} \sqrt{x + 2}$

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Solution

${lim}_{x \to \infty} {\sqrt{x + 1} - \sqrt{x}} \sqrt{x + 2}$

$= {lim}_{x \to \infty} [\sqrt{x + 1} - \sqrt{x}] \frac{[\sqrt{x + 1} + \sqrt{x}]}{[\sqrt{x + 1} + \sqrt{x}]} \times \frac{\sqrt{x + 2} \times \sqrt{x + 2}}{\sqrt{x + 2}}$

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

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

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

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

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

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limx→∞{√x+1−√x}√x+2

${lim}_{x \to \infty} {\sqrt{x + 1} - \sqrt{x}} \sqrt{x + 2}$