If $$f\left( x \right) =\sqrt { 1-\sqrt { 1-{ x }^{ 2 } } } $$, then $$f(x)$$ is

differentiable on $$(-1,0)\cup (0,1)$$

MCQ Questions for Class 11 Commerce Applied Mathematics Limits And Continuity Quiz 12

$\lim _{ x\rightarrow \infty }{ \frac { \sin { x } \sin { (\frac { \pi }{ 3 } +x) } \sin { (\frac { \pi }{ 3 } -x) } }{ x } } =$

0%
$\frac { 3 }{ 4 }$
0%
$\frac { 1 }{ 4 }$
0%
$\frac { 4 }{ 3 }$
0%
$0$

$\alpha , \beta , \ in (-\frac{\pi}{2},0)$ such that

$(sin \alpha +sin \beta ) +\frac{sin \alpha }{sin \beta} =0$ and

$(sin \alpha +sin \beta ) \frac{sin \alpha}{sin \beta }=-1$ and

$\lambda =\begin{matrix} lim \\ n\rightarrow \infty \end{matrix}\frac { 1+(2sin\quad theta\quad ){ }^{ 2n } }{ (2sin\quad \quad beta\quad ){ }^{ 2n } }$ then

0%
$2 \alpha +3 \beta = \frac{-5 \pi }{6}$
0%
$\lambda \pi +\alpha +\beta =\frac{5 \pi}{6}$
0%
$\alpha - \beta = \frac{ \pi }{3}$
0%
$\alpha + \beta = \frac{- \pi }{3}$

$\lim _{ x\rightarrow 0 }{ \frac { { 27 }^{ x }-{ 9 }^{ x }-{ 3 }^{ x }+1 }{ \sqrt { 2 } -\sqrt { 1+\cos { x } } } = }$

0%
$0$
0%
$8\sqrt { 2 } { (\log { 3 } ) }^{ 2 }$
0%
$8{ (\log { 3 } ) }^{ 2 }$
0%
$1$

$\lim _{ x\rightarrow }{ 0 } \{ (sinx-x)/{ x }^{ 3 })\}$ equals:

0%
$1/3$
0%
$-1/3$
0%
$1/6$
0%
$-1/6$

$\lim _{ x\rightarrow 0 }{ \frac { 1 }{ { x }^{ 8 } } [1-\cos { (\frac { { x }^{ 2 } }{ 2 } ) } ] } [1-\cos { (\frac { { x }^{ 2 } }{ 4 } ) } ]$

0%
$\frac { 1 }{ 8 }$
0%
$\frac { 1 }{ { 8 }^{ 2 } }$
0%
$\frac { 1 }{ { 8 }^{ 3 } }$
0%
$\frac { 1 }{ { 8 }^{ 4 } }$

$\lim _{x \rightarrow a}\left(2-\frac{a}{x}\right)^{\tan \left(\frac{\pi x}{2 a}\right)}$

0%
$e^{-\frac{a}{\pi}}$
0%
$e^{-\frac{2 a}{\pi}}$
0%
$e^{-\frac{2}{x}}$
0%
1

Solve

$\underset { x\rightarrow 0 }{ Lim } \frac { 8 }{ { x }^{ 8 } } \left( 1-cos\frac { { x }^{ 2 } }{ 2 } -cos\frac { { x }^{ 2 } }{ 4 } +cos\cfrac { { x }^{ 2 } }{ 2 } .cos\cfrac { { x }^{ 2 } }{ 4 } \right) =$

0%
$\cfrac { 1 }{ 16 }$
0%
$\cfrac { 1 }{ 15 }$
0%
$\cfrac { 1 }{ 32 }$
0%
$1$

$f\left( x \right) =\sqrt { 1-\sqrt { 1-{ x }^{ 2 } } }$ , then

$f(x)$ is

0%
continuous on $[-1,1]$
0%
differentiable on $(-1,0)\cup (0,1)$
0%
both (a) and (b)
0%
None of the above

$\underset { x\rightarrow 0 }{ lim } \cfrac { 8 }{ { x }^{ 8 } } \left( 1-cos\cfrac { { x }^{ 2 } }{ 2 } -cos\cfrac { { x }^{ 2 } }{ 4 } +cos\cfrac { { x }^{ 2 } }{ 2 } .cos\cfrac { { x }^{ 2 } }{ 4 } \right) =$

0%
$\cfrac { 1 }{ 16 }$
0%
$\cfrac { 1 }{ 15 }$
0%
$\cfrac { 1 }{ 32 }$
0%
$1$

$\displaystyle \lim _{ x\rightarrow 0 }{ \left[ \dfrac { 100\tan { x } .\sin { x } }{ { x }^{ 2 } } \right] }$ where

$[.]$ represents greatest integer function is

0%
$99$
0%
$100$
0%
$0$
0%
$98$

The value of

$\lim _ { x \rightarrow \dfrac { 1 } { 2 } } \dfrac { 2 \sin ^ { - 1 } x - \dfrac { \pi } { 2 } } { 1 - 2 x ^ { 2 } }$ is equal to

0%
$-1$
0%
$0$
0%
$1$
0%
$2$

$f(x) = \log_{1 - 2x}(1 + 2x)$ for

$x \ne 0$

$= k$ for

$x = 0$
is continuous at

$x = 0$ , find

$k.$

0%
$1$
0%
$-1$
0%
$0$
0%
$\frac{1}{2}$

The value of

$\lim _{ x\rightarrow \frac { \pi }{ 2 } }{ \tan ^{ 2 }{ \left( \sqrt { 2\sin ^{ 2 }{ x } +3\sin { x } +4 } -\sqrt { \sin ^{ 2 }{ x } +6\sin { x } +2 } \right) } }$ is equal to

0%
0
0%
$\cfrac{1}{11}$
0%
$\cfrac{1}{12}$
0%
$\cfrac{1}{8}$

$\mathop {\lim }\limits_{x \to \infty } \left( {\dfrac{{{x^2}\sin \left( {\dfrac{1}{x}} \right) - x}}{{1 - \left| x \right|}}} \right) =$

0%
0
0%
1
0%
-1`
0%
2

$\lim _{x \rightarrow 0}\left(\frac{e^{x}+e^{-x}-2}{x^{2}}\right)^{1 / x^{2}}$

0%
$e^{1 / 2}$
0%
$e^{1 / 4}$
0%
$e^{1 / 3}$
0%
$e^{1 / 12}$

The value of

$\displaystyle \lim _{ x\rightarrow 0 }\log_{\cos{2x}}{\cos{x}}+\log_{\cos{2x}}{\cos{2x}}$ equals

0%
$\dfrac{5}{4}$
0%
$\dfrac{17}{4}$
0%
$\dfrac{13}{16}$
0%
$\dfrac{29}{10}$

$\begin{matrix} lim \\ x\rightarrow 0 \end{matrix}(cos\quad +\quad sinx{ ) }^{ 1/x }$ is equal to

0%
e
0%
${ e }^{ 2 }$
0%
${ e }^{ -1 }$
0%
1

$\lim _{ x\rightarrow 0 }{ \frac { \sin { [\cos { x } ] } }{ 1+[\cos { x } ] } }$ is (where [] is G.I.F)

0%
1
0%
0
0%
does not exist
0%
2

$\underset { x\rightarrow a }{ lim } \left( \sin { \dfrac { x-a }{ 2 } \tan { \dfrac { \pi x }{ 2a } } } \right)$

0%
$a/\pi$
0%
$-a/\pi$
0%
$\pi/a$
0%
$-\pi/a$

Let

$a \in \left( 0 , \frac { \pi } { 2 } \right)$ , then the value of

$\lim _ { a \rightarrow 0 } \frac { 1 } { a ^ { 3 } } \int _ { 0 } ^ { a } \ell n (1+tan a tan x)dx$ is equal to

0%
$\frac { 1 }{ 3 }$
0%
$\frac { 1 }{ 2 }$
0%
$\frac { 1 }{ 6 }$
0%
1

$\lim _ { x \rightarrow 0 } \frac { \sin x } { x } = y$

0%
$y > 1$
0%
$y < 1$
0%
$y \geq 1$
0%
$y \leq 1$

The value of

$\mathop {{\text{Limit}}}\limits_{x \to 0} \frac{{\cos \left( {\sin x} \right) - \cos x}}{{{x^4}}}$ is equal to

0%
1/5
0%
1/6
0%
1/4
0%
1/2

$\lim_{x\rightarrow 0}\frac{1-cos^{3}x}{x sib x cos x }$ is equal to

0%
2/5
0%
3/5
0%
3/2
0%
3/4

The value of

$\lim _ { x \rightarrow 0 } \dfrac { \sin^3 ( \sqrt { x } ) \ln ( 1 + 3 x ) } { \left( \tan ^ { - 1 } \sqrt { x } \right) ^ { 2 } \left( e ^ { 5 ( \sqrt { x } ) } - 1 \right)x }$ is equal to

0%
$\frac { 1 } { 5 }$
0%
$\frac { 3 } { 5 }$
0%
$\frac { 2 } { 5 }$
0%
$\frac { 4 } { 5 }$

$\underset {x\rightarrow 0}{Lim} \frac {sin x}{x} = y$

0%
$y>1$
0%
$y<1$
0%
$y\ge 1$
0%
$y \le 1$

$\mathop {Lt}\limits_{x \to 1} {\left( {1 - x} \right)^{\tan \pi x}} =$

0%
$-1$
0%
$0$
0%
$1$
0%
$2$

$\begin{matrix} lim \\ x\rightarrow a \end{matrix}(2-\frac { a }{ x } ){ }^{ tan(\frac { \pi x }{ 2a } ) }$ is equal to

0%
$e^{-\frac{a}{\pi}}$
0%
$e^{-\frac{2a}{\pi}}$
0%
$e^{-\frac{2}{\pi}}$
0%
1

$\lim _{x \rightarrow0}\left(\frac{e^{x}+e^{-x}-2}{x^{2}}\right)^{1 / x^{2}}$ is

0%
$e^{1 / 2}$
0%
$e^{1 / 4}$
0%
$e^{1 / 3}$
0%
$e^{1 / 12}$

Let x be an irrational, then

$\underset { m\rightarrow \infty }{ lim } \underset { n\rightarrow \infty }{ lim } \left\{ cos(n!\pi x) \right\} ^{ 2m }$ equals

0%
0
0%
-1
0%
1
0%
Indeterminate

$\underset { \rightarrow }{ Lim } \frac { 1-{ cos }^{ 2 }x }{ xsin2x }$

0%
1/2
0%
3/2
0%
3/4
0%
1/4

$\displaystyle x\xrightarrow { lim } 5\quad \left(\frac{\sqrt{1-\cos(2x-10)}}{\sin (x-5)} \right)$

0%
$-\sqrt{2}$
0%
$\sqrt{2}$
0%
does not exist
0%
none of these

The value of

$\mathop {\lim }\limits_{x \to 0} \frac{{\sec 5x - \sec 3x}}{{\sec 3x - \sec x}}$

0%
-2
0%
1
0%
2
0%
-1/2

$\lim _ { x \rightarrow 0 } \int _ { 0 } ^ { x } \dfrac { \left( \tan ^ { - 1 } t \right) ^ { 2 } } { \sqrt { 1 + x ^ { 2 } } } d t$ is equal to

0%
$\pi ^ { 2 }$
0%
$\dfrac { \pi ^ { 2 } } { 2 }$
0%
$\dfrac { \pi ^ { 2 } } { 4 }$
0%
None of these

the value of

$\underset { x\longrightarrow \infty }{ lim } \frac { { X }^{ 4 }sin\left( \frac { 1 }{ x } \right) +{ x }^{ 3 } }{ 1+\left| x \right| ^{ 3 } }$

0%
1
0%
-1
0%
2
0%
does not exist

$\underset { x\rightarrow 1 }{ lim } { \left[ cosec { \dfrac { \pi x }{ 2 } } \right] }^{ { 1 }/{ \left( 1-x \right) } }$ (where

$[.]$ represents the greatest integer function) is equal to

0%
$0$
0%
$1$
0%
$\infty$
0%
$Does \ not \ exist$

f(X)=|x|+|x-1| is continuous at

0%
'0' only
0%
0,1 only
0%
Every where
0%
No where

Explanation

$\textbf{Step-1: Finding values of}$

$\mathbf{f(x)}$

$f(x)=|x|+|x-1|$

$f(x)=-x-|x-1|$

$\text{at}$

$x\leq{0}$

$=x-(x-1)$

$\text{at}$

$0\leq{x}<1$

$=x+(x-1)$

$\text{at}$

$x\geq{1}$

$\text{Hence, we get}$

$f(x)=1-2x$

$\text{at}$

$x\leq{0}$

$=1$

$\text{at}$

$0\leq{x}<1$

$=2x-1$

$\text{at}$

$x\geq{1}$

$\textbf{Step-2: Finding limits of the function at constraint values}$

$\text{At}$

$x=0,$

$\lim_{x\to 0} f(x)=1$

$\text{Hence,}$

$f(x)$

$\text{is continuous at}$

$x=0$

$\text{At}$

$x=1,$

$\lim_{x\to 1} f(x)=1$

$\text{Hence,}$

$f(x)$

$\text{is continuous at}$

$x=1$

$\text{Hence,}$

$f(x)$

$\text{is continuous everywhere}$

$\textbf{Hence,Correct option is (C)}$

Find:

$\underset { x\rightarrow 0 }{ lim } \quad \dfrac { 1-cos^{ 3 }x }{ xsin2x } =\quad$

0%
1/2
0%
3/2
0%
3/4
0%
1/4

$\displaystyle \lim_{x\rightarrow 0}\dfrac {\sin x - x}{x^{3}}$ is equal to

0%
$-\dfrac {1}{6}$
0%
$-\dfrac {1}{3}$
0%
$-\dfrac {1}{2}$
0%
$-1$

$\lim _{ x\rightarrow 0 }{ \dfrac { 1-\cos { x } }{ { { x\log { (1+x) } } } } }$ =

0%
1
0%
0
0%
-1
0%
1/2

$\alpha$ and

$\beta$ be the roots of the equation

$ax^{2} + bx + c = 0$ then

$\displaystyle \lim_{x\rightarrow \dfrac {1}{\alpha}} \sqrt {\dfrac {1 - \cos^{2} (cx^{2} + bx + a)}{4(1 - \alpha x)^{2}}}$

0%
Does not exist
0%
Equals $\left |\dfrac {c}{2\alpha} \left (\dfrac {1}{\alpha} +\dfrac {1}{\beta}\right )\right |$
0%
Equals $\left |\dfrac {c}{2\alpha} \left (\dfrac {1}{\alpha} - \dfrac {1}{\beta}\right )\right |$
0%
Equals $\left |\dfrac {c}{2} \left (\dfrac {1}{\alpha} +\dfrac {1}{\beta}\right )\right |$

The value of

$\begin{matrix} lim \\ x\rightarrow \frac { 1 }{ \sqrt { 2 } } \end{matrix}\dfrac { x-cos\left( { sin }^{ -1 }x \right) }{ 1-tan\left( { sin }^{ -1 }x \right) } is$

0%
$-\dfrac { 1 }{ \sqrt { 2 } }$
0%
$\dfrac { 1 }{ \sqrt { 2 } }$
0%
$\sqrt { 2 }$
0%
$-\sqrt { 2 }$

$\overset {lim}{x \rightarrow \pi/2} \dfrac{\sin(x \ cos x)}{cos(x\, \ sin x)}$ is equal to

0%
$1$
0%
$\dfrac{\pi}{2}$
0%
$\dfrac{2}{\pi}$
0%
does not exist

$\begin{matrix} lim \\ x\rightarrow 0 \end{matrix}\frac { xcot(4x) }{ { sin }^{ 2 }x{ cot }^{ 2 }\left( 2x \right) }$ is equal to

0%
0
0%
2
0%
1
0%
4

$\underset { x\rightarrow 0 }{ lim } \dfrac { sec4x-sec2x }{ sec3x-secx } =$

0%
3/2
0%
2/3
0%
1/3
0%
3/4

The value of

$\underset{x\rightarrow 1}{lim}(2-x)^{tan\left(\dfrac{\pi x}{2}\right)}$ is

0%
$e^{-2/pi}$
0%
$e^{1/pi}$
0%
$e^{2/pi}$
0%
$e^{-1/pi}$

$\underset{x \rightarrow 1} {lim}\dfrac{x^2-1}{\sin^2x+\cos x\cos (x+2)-\cos^2(x+1)}$ is-

0%
$0$
0%
$\dfrac{1}{\cos 1}$
0%
$\dfrac{2}{\sin 2}$
0%
$\dfrac{1}{2\cos 1}$

$\lim_{x\rightarrow 1}\frac{1-x^{-2/3}}{1-x^{-1/3}}$

0%
2
0%
1
0%
0
0%
does not exist

$lim_{n \to \infty}\dfrac{-3n + (-1)^n}{4n - (-1)^n}$ is equal to

0%
$$-\dfrac{3]{4}$$
0%
o if n is even
0%
$-\dfrac{3}{4}$ if n is odd
0%
None of these

$\displaystyle \lim _{x \rightarrow 1}\left(\dfrac{x^{4}+x^{2}+x+1}{x^{2}-x+1}\right)^{\dfrac{1-\cos (x+1)}{(x+1)^{2}}}$ is equal to:

0%
1
0%
$(2 / 3)^{1 / 2}$
0%
$(3 / 2)^{1 / 2}$
0%
$e^{1 / 2}$

$\displaystyle \lim _{x \rightarrow 0}\left(\dfrac{1^{x}+2^{x}+3^{x}+\cdots+n^{x}}{n}\right)^{1 / x}$ is equal to

0%
$(n !)^{n}$
0%
$(n !)^{1 / n}$
0%
$n !$
0%
$\ln (n !)$

CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity Quiz 12 - MCQExams.com

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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity Quiz 12 - MCQExams.com

0:0:1

Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

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