Lim x -10-7-2 x-5-2-x2-10

Lim_x →√(10)√(7+2x) (√(5)+√(2))/x^2 10 =lim_x →√(10)√(7+2x) (√((√(5)+√(2))^2))/x^2 10 =lim_x →√(10)√(7+2x) (√(7+2√(10)))/x^2 10 Rationalising the numerator =l ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x →√10√7+...

Question

${lim}_{x \to \sqrt{10}} \frac{\sqrt{7 + 2 x} - (\sqrt{5} + \sqrt{2})}{x^{2} - 10}$

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Solution

${lim}_{x \to \sqrt{10}} \frac{\sqrt{7 + 2 x} - (\sqrt{5} + \sqrt{2})}{x^{2} - 10}$

$= {lim}_{x \to \sqrt{10}} \frac{\sqrt{7 + 2 x} - (\sqrt{(\sqrt{5} + \sqrt{2})^{2}})}{x^{2} - 10}$

$= {lim}_{x \to \sqrt{10}} \frac{\sqrt{7 + 2 x} - (\sqrt{7 + 2 \sqrt{10}})}{x^{2} - 10}$

Rationalising the numerator

$= {lim}_{x \to \sqrt{10}} \frac{\sqrt{7 + 2 x} - (\sqrt{7 + 2} \sqrt{10})}{x^{2} - 10} \times \frac{\sqrt{7 + 2 x} + (\sqrt{7 + 2 \sqrt{10}})}{\sqrt{7 + 2 x} + (\sqrt{\sqrt{7 + 2 \sqrt{10}}})}$

$= {lim}_{x \to \sqrt{10}} \frac{7 + 2 x - 7 - 2 \sqrt{10}}{(x^{2} - 10) (\sqrt{7 + 2 x} + (\sqrt{7 + 2 \sqrt{10}}))}$

$= {lim}_{x \to \sqrt{10}} \frac{2 (x - \sqrt{10})}{(x^{2} - 10) (\sqrt{7 + 2 x} + (\sqrt{7 + 2 \sqrt{10}}))}$

$= {lim}_{x \to \sqrt{10}} \frac{2}{(x + \sqrt{10}) (\sqrt{7 + 2 x} + (\sqrt{7 + 2 \sqrt{10}}))}$

$= \frac{2}{(\sqrt{10} + \sqrt{10}) (\sqrt{7 + 2 \sqrt{10}} + (\sqrt{7 + 2 \sqrt{10}}))}$

$= \frac{2}{(2 \sqrt{10}) (2 \sqrt{7 + 2 \sqrt{10}})}$

$= \frac{1}{(2 \sqrt{10}) (2 \sqrt{7 + 2 \sqrt{10}})}$

$= \frac{1}{(2 \sqrt{10}) (\sqrt{5} + \sqrt{2})}$

$= \frac{(\sqrt{5} - \sqrt{2})}{(6 \sqrt{10})}$

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Similar questions

Q. Find :

lim x \to \sqrt{10} \frac{\sqrt{7 + 2 x} - (\sqrt{5} + \sqrt{2})}{x^{2} - 10}

Evaluate $l i m x \to \sqrt{10} \frac{\sqrt{7 - 2 x} - (\sqrt{5} - \sqrt{2})}{x^{2} - 10}$

Differentiate $x^{2} c o s x$ by first principle.

l i m x \to \sqrt{10} \frac{\sqrt{7 + 2 x} - (\sqrt{5} + \sqrt{2})}{x^{2} - 10}

is equal to -

\lim_{x \to \sqrt{10}} \frac{\sqrt{7 + 2 x} - (\sqrt{5} + \sqrt{2})}{x^{2} - 10}

\frac{x^{2} - 7 x + 10}{x^{2} - 5 x - 14} \times \frac{x^{3} + 8}{x^{3} - 8} \times \frac{x^{2} + 2 x + 4}{x^{2} - 2 x + 4} \times \frac{x - 7}{x - 5} = ?

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limx→√10√7+2x−(√5+√2)x2−10

${lim}_{x \to \sqrt{10}} \frac{\sqrt{7 + 2 x} - (\sqrt{5} + \sqrt{2})}{x^{2} - 10}$