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12. limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS12. limits1 Mark
MCQ
If $f(x) = \left\{ \begin{array}{l}\frac{{\sin [x]}}{{[x]}},{\rm{ when\,\, }}[x] \ne 0\\\,\,\,\,\,\,\,\,\,0,{\rm{ when \,\,}}[x] = 0\end{array} \right.$ where $[x]$ is greatest integer function, then $\mathop {\lim }\limits_{x \to 0} f(x) = $
  • A
    $-1$
  • B
    $1$
  • C
    $0$
  • ✓
    None of these
✓

Answer

Correct option: D.
None of these
d
(d) In closed interval of $x = 0$ at right hand side $[x] = 0$ and at left hand side $[x] = - 1.$ Also $[0]=0.$ 

Therefore function is defined as $f(x) = \left\{ \begin{array}{l}\frac{{\sin \,[x]}}{{[x]}}\,\,( - 1 \le x < 0)\\\;\;\;\;\;0\;\;(0 \le x < 1)\end{array} \right.$

$\therefore$ Left hand limit $ = \mathop {\lim }\limits_{x \to 0 - } f(x) = \mathop {\lim }\limits_{x \to 0 - } \,\frac{{\sin \,[x]}}{{[x]}}$

$ = \frac{{\sin \,( - 1)}}{{ - 1}} = \sin {1^c}$

Right hand limit $= 0$. 

Hence limit doesn't exist.

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