Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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If F ( X ) = { X Sin 1 X , X ≠ 0 0 , X = 0 , Then Lim X → 0 F ( X ) Equals - Mathematics

MCQ

If \[f\left( x \right) = \left\{ \begin{array}{l}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x = 0\end{array}, \right.\] then \[\lim_{x \to 0} f\left( x \right)\]  equals 

Options

  •  1 

  •  0 

  •  −1 

  •  none of these 

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Solution

 0 

\[f\left( x \right) = \binom{x\sin\left( \frac{1}{x} \right), x \neq 0}{0, x = 0}\] 

LHL: 

\[\lim_{x \to 0^-} f\left( x \right)\]
\[ = \lim_{x \to 0^-} \left[ x\sin\left( \frac{1}{x} \right) \right]\]


Let x = 0 – h, where h → 0. 

\[= \lim_{h \to 0} \left[ \left( - h \right) \times \sin\left( - \frac{1}{h} \right) \right]\] 


= 0 × The oscillating number between –1 and 1
= 0
RHL 

\[\lim_{x \to 0^+} f\left( x \right)\] 

Let x = 0 + h, where h → 0.

\[= \lim_{h \to 0} \left[ h \times \sin\left( \frac{1}{h} \right) \right]\] 


= 0 × The oscillating number between –1 and 1
= 0
LHL = RHL = 0 

\[\therefore \lim_{x \to 0} f\left( x \right) = 0\]

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

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Q 24 | Page 79
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