Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# 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 $\lim_{x \to 3^-} \frac{x}{\left[ x \right]} .$

#### 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