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

Let F ( X ) = { X + 1 , I F X ≥ 0 X − 1 , I F X < 0 . Prove that Lim X → 0 F ( X ) Does Not Exist. - Mathematics

Let $f\left( x \right) = \left\{ \begin{array}{l}x + 1, & if x \geq 0 \\ x - 1, & if x < 0\end{array} . \right.$Prove that $\lim_{x \to 0} f\left( x \right)$ does not exist.

Solution

$f\left( x \right) = \begin{cases}x + 1, & x \geq 0 \\ x - 1, & x < 0\end{cases}$
$\text{ RHL }:$
$\lim_{x \to 0^+} f\left( x \right)$
$= \lim_{x \to 0} \left( x + 1 \right)$
$\text{ Let } x = 0 + h, \text{ where } h \to 0 .$
$\lim_{h \to 0} \left( 0 + h + 1 \right)$
$= 1$
$\text{ LHL }:$
$\lim_{x \to 0^-} f\left( x \right)$
$= \lim_{x \to 0^-} \left( x - 1 \right)$
$\text{ Let } x = 0 - h, \text{ where } h \to 0 .$
$\lim_{h \to 0} \left( 0 - h - 1 \right)$
$= - 1$
$LHL \neq RHL$
$\text{ Thus }, \lim_{x \to 0} f\left( x \right) \text{ does not exist } .$

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.1 | Q 5 | Page 11