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

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
$\mathop {\lim }\limits_{x \to a} f(x) = a$
✓
$f(x)$ is continuous at $x = a$
C
$f(x)$ is discontinuous at $x = a$
D
None of these

✓

Answer

Correct option: B.

$f(x)$ is continuous at $x = a$

b
(b) $f(a) = 0$

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

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 $x = a$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 5. continuity and differentiation→5. continuity and differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The area common to the ellipse $\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$ and $\frac{\text{x}^2}{\text{b}^2}+\frac{\text{y}^2}{\text{a}^2}=1,0<\text{b}<\text{a}$ is:

$(\text{a}+\text{b})^2\tan^{-1}\frac{\text{b}}{\text{a}}$
$(\text{a}+\text{b})^2\tan^{-1}\frac{\text{a}}{\text{b}}$
$4\text{a}+\text{b}\tan^{-1}\frac{\text{b}}{\text{a}}$
$4\text{a}+\text{b}\tan^{-1}\frac{\text{a}}{\text{b}}$

If $A_1B_1C_1,\, A_2B_2C_2,\, A_3B_3C_3$ are three digit number each of which is divisible by $k$ and $\Delta = \left| {\begin{array}{*{20}{c}}
{{A_1}{\kern 1pt} }&{{B_1}}&{{C_1}} \\
{{A_2}}&{{B_2}}&{{C_2}} \\
{{A_3}}&{{B_3}}&{{C_3}}
\end{array}} \right|$ ; then $\Delta $ is divisible by

Evaluate: $\int\sec^2(7-4\text{x})\text{dx}.$

$-\frac{1}{4}\tan(7-4\text{x})+\text{c}$
$\frac{1}{4}\tan(7-4\text{x})+\text{c}$
$\frac{1}{4}\tan(7+4\text{x})+\text{c}$
$-\frac{1}{4}\tan(7\text{x}-4)+\text{c}$

If AD, BE and CF are $\triangle\text{ABC},$ then $\vec{\text{AD}}+\vec{\text{BE}}+\vec{\text{CF}}$

$\vec{0}$
1
0
2

If $y=y(x)$ is the solution of the equaiton $e ^{\sin y} \cos y \frac{ dy }{ dx }+ e ^{\sin y} \cos x =\cos x , y (0)=0$ then $1+ y \left(\frac{\pi}{6}\right)+\frac{\sqrt{3}}{2} y \left(\frac{\pi}{3}\right)+\frac{1}{\sqrt{2}} y \left(\frac{\pi}{4}\right)$ is equal to

The solution of $\frac{{dy}}{{dx}} + \sqrt {\,\left( {\frac{{1 - {y^2}}}{{1 - {x^2}}}} \right)} \, = \,0$ is

The equation $\sin^{-1}\text{x}-\cos^{-1}\text{x}=\cos^{-1}(\frac{\sqrt3}{2})$ has:

Nique solution.
No solution.
Infinitely many solution.
None of these.

$\int {\frac{{dx}}{{{{\cos }^3}x\sqrt {2\sin 2x} }}} $ is equal to

If $x + y + z = 0,|x| = |y| = |z| = 2$ and $\theta $ is angle between $y$ and $z$, then the value of ${\rm{cose}}{{\rm{c}}^2}\theta + {\cot ^2}\theta $ is equal to

A value of $x$ for which $\sin \,\left( {{{\cot }^{ - 1}}\,\left( {1 + x} \right)} \right) = \cos \,\left( {{{\tan }^{ - 1}}\,x} \right)$, is