Lim x -2-3-2 x-2-1-x2-2

Lim_x →√(2)√(3+2x) (√(2)+1)/x^2 2 =lim_x →√(2)√(3+2x) (√((√(2)+1)^2))/x^2 2 =lim_x →√(2)√(3+2x) (√(3+2√(2)))/x^2 2 =lim_x →√(2)√(3+2x) (√(3+2√(2)))/x^2 2×√(3+2x ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x →√2√3+2...

Question

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

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Solution

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

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

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

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

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

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

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

$= \frac{2}{(\sqrt{2} + \sqrt{2}) (\sqrt{3 + 2 \sqrt{2}} + (\sqrt{3 + 2 \sqrt{2}}))}$

$= \frac{2}{2 \sqrt{2} (2 \sqrt{3} + 2 \sqrt{2})}$

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

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

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Similar questions

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lim x \to \sqrt{2} \frac{\sqrt{3 + 2 x} - (2 \sqrt{2} + 3)}{x^{2} - 2}

Q. Find the limits :

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lim x \to 1 [\frac{x^{2} + 1}{x + 100}]

i i .

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

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i v .

lim x \to 2 [\frac{x^{3} - 2 x^{2}}{x^{2} - 5 x + 6}]

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limx→√2√3+2x−(√2+1)x2−2

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