Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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Find Lim X → 3 + X [ X ] . is It Equal to Lim X → 3 − X [ X ] . - Mathematics

Find \[\lim_{x \to 3^+} \frac{x}{\left[ x \right]} .\]  Is it equal to \[\lim_{x \to 3^-} \frac{x}{\left[ x \right]} .\]

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Solution

\[\lim_{x \to 3^+} \frac{x}{\left[ x \right]}\]
\[\text{ Let } x = 3 + \text{ h, where h } \to 0 . \]
\[ \Rightarrow \lim_{h \to 0} \left( \frac{3 + h}{\left[ 3 + h \right]} \right) = \frac{3}{3} = 1\]
\[Also, \lim_{x \to 3^-} \left( \frac{x}{\left[ x \right]} \right)\]
\[\text{ Let } x = 3 - \text{ h, where h } \to 0 . \]
\[ \lim_{h \to 0} \left( \frac{3 - h}{\left[ 3 - h \right]} \right)\]
\[ = \frac{3}{2}\]
\[ \therefore \lim_{x \to 3^-} \frac{x}{\left[ x \right]} \neq \lim_{x \to 3^+} \frac{x}{\left[ x \right]}\] 

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.1 | Q 18 | Page 12
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