Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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Prove that F ( X ) = { X − | X | X , X ≠ 0 2 , X = 0 is Discontinuous at X = 0 - Mathematics

Sum

Prove that  \[f\left( x \right) = \begin{cases}\frac{x - \left| x \right|}{x}, & x \neq 0 \\ 2 , & x = 0\end{cases}\] is discontinuous at x = 0

 

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Solution

The given function can be rewritten as 

\[f\left( x \right) = \begin{cases}\frac{x - x}{x}, \text{ when } x > 0 \\ \frac{x + x}{x}, \text{ when } x < 0 \\ 2, \text{ when } x = 0\end{cases}\]
\[\Rightarrow\]  \[f\left( x \right) = \begin{cases}0, \text{ when } x > 0 \\ 2, \text{ when }  x < 0 \\ 2, \text{ when } x = 0\end{cases}\]
We have
(LHL at x = 0) = 
\[\lim_{x \to 0^-} f\left( x \right) = \lim_{h \to 0} f\left( 0 - h \right) = \lim_{h \to 0} f\left( - h \right) = \lim_{h \to 0} 2 = 2\]
(RHL at x = 0) = 
\[\lim_{x \to 0^+} f\left( x \right) = \lim_{h \to 0} f\left( 0 + h \right) = \lim_{h \to 0} f\left( h \right) = \lim_{h \to 0} 0 = 0\]
∴ ​ \[\lim_{x \to 0^-} f\left( x \right) \neq \lim_{x \to 0^+} f\left( x \right)\]

Thus,  f(x) is discontinuous at x = 0.

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 9 Continuity
Exercise 9.1 | Q 40 | Page 21
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