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STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
MCQ
If $f(x) = \left\{ \begin{array}{l}({x^2}/a) - a,\;\;{\rm{when}}\;x < a\\\;\;\;\;\;\;\;\;\;\;\;0,\;\;{\rm{when}}\;x = a{\rm{,}}\\a - ({x^2}/a),\;\;{\rm{when\,\, }}x > a\end{array} \right.$ then
  • A
    $\mathop {\lim }\limits_{x \to a} f(x) = a$
  • B
    $f(x)$ is continuous at $x = a$
  • C
    $f(x)$ is discontinuous at $x = a$
  • D
    None of these
✓

Answer

$f(a) = 0$
$\mathop {\lim }\limits_{x \to a - } \,f(x) = \mathop {\lim }\limits_{x \to a - } \left( {\frac{{{x^2}}}{a} - a} \right)$
$= \mathop {\lim }\limits_{h \to 0} \,\left\{ {\frac{{{{(a - h)}^2}}}{a} - a} \right\}$
$= 0$
and $\mathop {\lim }\limits_{x \to a + } f(x)$
$= \mathop {\lim }\limits_{h \to 0} \,\,\left\{ {a - \frac{{{{(a + h)}^2}}}{a}} \right\}$
$= 0$
Hence it is continuous at $x = a.$

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