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

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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
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