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

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

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?

APPEARS IN

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