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

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,\,\,\frac{{1 - \cos 4x}}{{{x^2}}},\;\;{\rm{when}}\,x < 0\\\,\,\,\,\,\,\,\,\,\,\,a,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{when}}\,\,x = 0\\\frac{{\sqrt x }}{{\sqrt {(16 + \sqrt x )} - 4}},\,\,{\rm{when}}\,\, x > 0\end{array} \right.$, is continuous at $x = 0$, then the value of $'a'$ will be
  • ✓
    $8$
  • B
    $-8$
  • C
    $4$
  • D
    None of these
✓

Answer

Correct option: A.
$8$
a
(a) $\mathop {\lim }\limits_{x \to 0 - } f(x) = \mathop {\lim }\limits_{x \to 0 - } \left( {\frac{{2\,{{\sin }^2}2x}}{{{{(2x)}^2}}}} \right)\,4 = 8$

$\mathop {\lim }\limits_{x \to 0 + } f(x) = \mathop {\lim }\limits_{x \to 0 + } \sqrt {16 + \sqrt x + 4} = 8$.

Hence $a = 8$.

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