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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
If $f(x)\, = \frac{{2 - \sqrt {x + 4} }}{{\sin 2x}},\,\,(x \ne 0),$ is continuous function at $x = 0$, then $f(0)$ equals
  • A
    $\frac{1}{4}$
  • B
    $ - \frac{1}{4}$
  • C
    $\frac{1}{8}$
  • ✓
    $ - \frac{1}{8}$
✓

Answer

Correct option: D.
$ - \frac{1}{8}$
d
(d) If $f(x)$ is continuous at $x = 0,$ then

$f(0)\, = \,\mathop {\lim }\limits_{x \to 0} f(x)$ $ = \mathop {\lim }\limits_{x \to 0} \frac{{2 - \sqrt {x + 4} }}{{\sin 2x}}$, $\left( {\frac{0}{0}{\rm{ }}\,{\rm{form}}} \right)$

Using $L -$ Hospital’s rule, 

$f(0) = \mathop {\lim }\limits_{x \to 0} \frac{{\left( { - \frac{1}{{2\sqrt {x + 4} }}} \right)}}{{2\cos 2x}} = - \frac{1}{8}$.

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