જો f(x) = l ( x^2 /a) - a, ; ; when ;x < a ; ; ; ; ; ; ; ; ; ; ;0… - Vidyadip

MCQ

જો $f(x) = \left\{ \begin{array}{l}({x^2}/a) - a,\;\;{\rm{when}}\;x < a\\\;\;\;\;\;\;\;\;\;\;\;0,\;\;{\rm{when}}\;x = a{\rm{,}}\\a - ({x^2}/a),\;\;{\rm{when \,\,}}x > a\end{array} \right.$ તો

A
$\mathop {\lim }\limits_{x \to a} f(x) = a$
✓
$f(x)$ એ $x = a$ આગળ સતત છે.
C
$f(x)$ એ $x = a$ આગળ અસતત છે.
D
એકપણ નહી.

✓

Answer

Correct option: B.

$f(x)$ એ $x = a$ આગળ સતત છે.

$f(a) = 0$
$\mathop {\lim }\limits_{x \to a - } \,f(x) = \mathop {\lim }\limits_{x \to a - } \left( {\frac{{{x^2}}}{a} - a} \right) = \mathop {\lim }\limits_{h \to 0} \,\left\{ {\frac{{{{(a - h)}^2}}}{a} - a} \right\} = 0$
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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