Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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Lim X → √ 6 √ 5 + 2 X − ( √ 3 + √ 2 ) X 2 − 6 - Mathematics

\[\lim_{x \to \sqrt{6}} \frac{\sqrt{5 + 2x} - \left( \sqrt{3} + \sqrt{2} \right)}{x^2 - 6}\] 

 

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Solution

\[\lim_{x \to \sqrt{6}} \left[ \frac{\sqrt{5 + 2x} - \left( \sqrt{3} + \sqrt{2} \right)}{x^2 - 6} \right]\] 

=  \[\lim_{x \to \sqrt{6}} \left[ \frac{\sqrt{5 + 2x} - \sqrt{\left( \sqrt{3} + \sqrt{2} \right)^2}}{x^2 - \left( \sqrt{6} \right)^2} \right]\]

=  \[\lim_{x \to \sqrt{6}} \left[ \frac{\sqrt{5 + 2x} - \sqrt{3 + 2 + 2\sqrt{6}}}{\left( x - \sqrt{6} \right)\left( x + \sqrt{6} \right)} \right]\] 

=  \[\lim_{x \to \sqrt{6}} \left[ \frac{\sqrt{5 + 2x} - \sqrt{5 + 2\sqrt{6}}}{\left( x - \sqrt{6} \right)\left( x + \sqrt{6} \right)} \right]\] 

Rationalising the numerator: 

\[\lim_{x \to \sqrt{6}} \left[ \frac{\left( \sqrt{5 + 2x} - \sqrt{5 + 2\sqrt{6}} \right)\left( \sqrt{5 + 2x} + \sqrt{5 + 2\sqrt{6}} \right)}{\left( x - \sqrt{6} \right)\left( x + \sqrt{6} \right)\left( \sqrt{5 + 2x} + \sqrt{5 + 2\sqrt{6}} \right)} \right]\] 

= \[\lim_{x \to \sqrt{6}} \left[ \frac{\left( 5 + 2x \right) - \left( 5 + 2\sqrt{6} \right)}{\left( x - \sqrt{6} \right)\left( x + \sqrt{6} \right)\left( \sqrt{5 + 2x} + \sqrt{5 + 2\sqrt{6}} \right)} \right]\] 

=  \[\lim_{x \to \sqrt{6}} \left[ \frac{2\left( x - \sqrt{6} \right)}{\left( x - \sqrt{6} \right)\left( x + \sqrt{6} \right)\left( \sqrt{5 + 2x} + \sqrt{5 + 2\sqrt{6}} \right)} \right]\] 

=  \[\frac{2}{\left( \sqrt{6} + \sqrt{6} \right)\left( \sqrt{5 + 2\sqrt{6}} + \sqrt{5 + 2\sqrt{6}} \right)}\]
= \[\frac{1}{2\sqrt{6}\left( \sqrt{\left( \sqrt{3} + \sqrt{2} \right)^2} \right)}\] 
= \[\frac{1}{2\sqrt{6}\left( \sqrt{3} + \sqrt{2} \right)}\] 

= \[\frac{1}{2\sqrt{6}\left( \sqrt{3} + \sqrt{2} \right)} \times \frac{\left( \sqrt{3} - \sqrt{2} \right)}{\left( \sqrt{3} - \sqrt{2} \right)}\] 

=  \[\frac{\sqrt{3} - \sqrt{2}}{2\sqrt{6}\left( 3 - 2 \right)}\] 
= \[\frac{\sqrt{3} - \sqrt{2}}{2\sqrt{6}}\]

 

 
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.4 | Q 33 | Page 29
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