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Model Paper 3 — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsModel Paper 35 Marks
Question
Evaluate : $\lim _{x \rightarrow \sqrt{10}} \frac{\sqrt{7-2 x}-(\sqrt{5}-\sqrt{2})}{x^2-10}$
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Answer

We have,
$\lim _{x \rightarrow \sqrt{10}} \frac{\sqrt{7-2 x}-(\sqrt{5}-\sqrt{2})}{x^2-10}$
$=\lim _{x \rightarrow \sqrt{10}} \frac{\sqrt{7-2 x}-\sqrt{(\sqrt{5}-\sqrt{2})^2}}{x^2-10}\left(\text { form } \frac{0}{0}\right)$
$=\lim _{x \rightarrow \sqrt{10}} \frac{\sqrt{7-2 x}-\sqrt{7-2 \sqrt{10}}}{x^2-10}\left(\text { form } \frac{0}{0}\right)$
$=\lim _{x \rightarrow \sqrt{10}} \frac{\sqrt{7-2 x}-\sqrt{7-2 \sqrt{10}}}{x^2-10} \times \frac{\sqrt{7-2 x}+\sqrt{7-2 \sqrt{10}}}{\sqrt{7-2 x}+\sqrt{7-2 \sqrt{10}}}$
$=\lim _{x \rightarrow \sqrt{10}} \frac{(7-2 x)-(7-2 \sqrt{10})}{(x-\sqrt{10})(x+\sqrt{10})\{\sqrt{7-2 x}+\sqrt{7-2} \sqrt{10}\}}$
$=\lim _{x \rightarrow \sqrt{10}} \frac{-2 x+2 \sqrt{10}}{(x-\sqrt{10})(x+\sqrt{10})\{\sqrt{7-2 x}+\sqrt{7-2} \sqrt{10}\}}$
$=\lim _{x \rightarrow \sqrt{10}} \frac{-2(x-\sqrt{10})}{(x-\sqrt{10})(x+\sqrt{10})\{\sqrt{7-2 x}+\sqrt{7-2 \sqrt{10}}\}}$
$=\lim _{x \rightarrow \sqrt{10}} \frac{-2}{(x+\sqrt{10})\{\sqrt{7-2 x}+\sqrt{7-2 \sqrt{10}}\}}$
$=\lim _{x \rightarrow \sqrt{10}} \frac{-2}{2 \sqrt{10}\{\sqrt{7-2 \sqrt{10}}+\sqrt{7-2 \sqrt{10}}\}}$
$=\frac{-1}{\sqrt{10} \times 2 \times \sqrt{7-2 \sqrt{10}}}=\frac{-1}{2 \sqrt{10}(\sqrt{5}-\sqrt{2})}\left[\because(\sqrt{5}-\sqrt{2})^2=7-2 \sqrt{10}\right]$
$=\frac{-1}{2 \sqrt{10}} \times \frac{(\sqrt{5}+\sqrt{2})}{3}$
$=-\frac{(\sqrt{5}+\sqrt{2})}{6 \sqrt{10}}$

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