Lim x -2x2-2-x2-2 x-4

Lim_x →√(2)x^2 2/x^2+√(2x) 4 =lim_x →√(2)(x)^2 (√(2))^2/x^2+√(2x) 4 It is of the form 0/0 =lim_x →√(2)(x √(2))(x+√(2))/x^2+2√(2x) √(2x) 4 =lim_x →√(2)(x √(2))( ...

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Byju's Answer

Standard XII

Mathematics

Factorization Method Form to Remove Indeterminate Form

lim x →√2x2-2...

Question

${lim}_{x \to \sqrt{2}} \frac{x^{2} - 2}{x^{2} + \sqrt{2 x} - 4}$

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Solution

${lim}_{x \to \sqrt{2}} \frac{x^{2} - 2}{x^{2} + \sqrt{2 x} - 4} = {lim}_{x \to \sqrt{2}} \frac{(x)^{2} - (\sqrt{2})^{2}}{x^{2} + \sqrt{2 x} - 4}$

It is of the form $\frac{0}{0}$

$= {lim}_{x \to \sqrt{2}} \frac{(x - \sqrt{2}) (x + \sqrt{2})}{x^{2} + 2 \sqrt{2 x} - \sqrt{2 x} - 4}$

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

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

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

$= \frac{2}{3}$

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limx→√2x2−2x2+√2x−4

limx→√2x2−2x2+√2x−4=limx→√2(x)2−(√2)2x2+√2x−4 It is of the form 00 =limx→√2(x−√2)(x+√2)x2+2√2x−√2x−4 =limx→√2(x−√2)(x+√2)x(x=2√2)−√2(x+2√2) =limx→√2(x−√2)(x+√2)(x+2√2)(x−√2) =√2+√2√2+2√2=2√23√2 =23

${lim}_{x \to \sqrt{2}} \frac{x^{2} - 2}{x^{2} + \sqrt{2 x} - 4}$