CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity     Quiz 9 - MCQExams.com

$$ \underset { n \rightarrow \infty }{ Lt } \sum _{ r=2n+1\quad  }^{ 3n } \dfrac {n}{r^2 - n^2} $$ is equal to :
  • $$ l n \sqrt{\frac {2}{3}} $$
  • $$ l n \sqrt{\frac {3}{2}} $$
  • $$ l n \frac {2}{3} $$
  • $$ l n \frac {3}{2} $$
The value of $$\displaystyle \lim_{x \rightarrow 1^{-}}\dfrac {1 - \sqrt {x}}{(\cos^{-1}x)^{2}}$$
  • $$\dfrac{-1}{4}$$
  • $$1/2$$
  • $$2$$
  • None of these
$$\displaystyle \lim_{I\rightarrow \left (\dfrac {\pi}{2}\right )} = \int_{0}^{t}\tan \theta \sqrt {\cos \theta} ln (\cos \theta) d\theta$$ is equal to
  • $$-4$$
  • $$4$$
  • $$-2$$
  • Does not exists
If $$f '$$ (0) = 0 and f(x) is a differentiable and increasing function,then lim $$ x \rightarrow 0$$  $$\frac {x.f ' (x^2)}{f ' (x)}$$
  • is always equal to zero
  • may not exist as left hand limit may not exist
  • may not exist as left hand limit may not exist
  • right hand limit is always zero
Consider $$f(x)=\lim _{ n\rightarrow \infty  }{ \cfrac { { x }^{ n }-\sin { { x }^{ n } }  }{ { x }^{ n }+\sin { { x }^{ n } }  }  } $$ for $$x>0,x\neq 1,f(1)=0$$ then
  • $$f$$ is continuous at $$x=1$$
  • $$f$$ has a discontinuity at $$x=1$$
  • $$f$$ has an infinite or oscillatory discontinuity at $$x=1$$
  • $$f$$ has a removal type of discontinuity at $$x=1$$
If $$\sum _{ r=1 }^{ k }{ \cos ^{ -1 }{ \beta  }  } =\cfrac { k\pi  }{ 2 } $$ for any $$k\ge 1$$ and $$A=\sum _{ r=1 }^{ k }{ { \left( { \beta  }_{ r } \right)  }^{ r } } $$, then $$\lim _{ x\leftarrow A }{ \cfrac { { \left( 1+x \right)  }^{ 1/3 }-{ \left( 1-2x \right)  }^{ 1/4 } }{ x+{ x }^{ 2 } }  } $$ is equal to
  • $$0$$
  • $$\cfrac{1}{2}$$
  • $$\cfrac { \pi }{ 2 } $$
  • $$\cfrac { 5 }{ 6 } $$
$$\displaystyle \lim_{x \rightarrow 0} \frac{ae^x + b cos x + c. e^{-x}}{sin^2 x} = 4$$ then b =
  • 2
  • 4
  • -2
  • -4
$$\lim_{x\rightarrow \dfrac{\pi}{6}} \dfrac{2 sin^{2} x + sin x - 1}{2 sin^{2} x - 3 sin x + 1}$$
  • 6
  • -6
  • -3
  • 3
Let f:R$$ \rightarrow  $$(0,1) be a continuous function.. Then, which of the following function(s) has (have) the value zero at some point in the interval (0,1)?
  • $$ e ^ { x } - \int _ { 0 } ^ { 1 } f ( t ) \sin t d t $$
  • $$ f ( x ) + \int _ { 0 } ^ { 1 } f ( t ) \sin t d t $$
  • $$ x - \int _ { 0 } ^ { \frac { \pi } { 2 } - x } f ( t ) \cos t d t $$
  • $$ x ^ { 3 } - f ( x ) $$
$$\mathop {\lim }\limits_{x \to {a^ + }} {{\left\{ x \right\}\sin \left( {x - a} \right)} \over {{{\left( {x - a} \right)}^2}}}$$ 

is equal to (where {.} denotes the fraction
part of x and $$a \in N$$
  • 0
  • 1
  • does not exist
  • none of thes
If $$f\left( x \right) = \left\{ \begin{array}{l}\frac{{1 - \left| x \right|}}{{1 + x}},{\rm{ }}x \ne  - 1\\1,{\rm{          }}x =  - 1{\rm{     }}\end{array} \right.$$   then $$f\left( {\left[ {2x} \right]} \right),$$ where $$\left[ {} \right]$$ represents the greatest integer function , is 
  • discontinuous at $$x = - 1$$
  • continuous at $$x = 0$$
  • continuous at $$x = \frac{1}{2}$$
  • continuous at $$x = 1$$
If $$f : R \rightarrow R$$ is defined by
$$f(x) = \left \{\begin{matrix} \dfrac{x + 2}{x^2 + 3x + 2} & if & x \in R - \{-1, -1\} \\ -1 & if & x = -2 \\ 0 & if  &x = -1\end{matrix} \right.$$ then $$f(x)$$ continuous on the set 
  • $$R$$
  • $$R - \{-2\}$$
  • $$R - \{-1, -2\}$$
  • $$R-\{-1\}$$
If $$l=\lim\limits_{n\to 3}\dfrac{x^2-9}{\sqrt{x^2+7}-4}$$ and $$m=\lim\limits_{n\to -3}\dfrac{x^2-9}{\sqrt{x^2+7}-4}$$, then
  • $$l\ne m$$
  • $$l=2m$$
  • $$l=-m$$
  • $$l=m$$
Let$$f(\theta) = \dfrac{1}{tan^{9}\theta} {(1+tan\theta)^{10}+(2+tan\theta)^{10}+....+(20+tan\theta)^{10}}-20tan\theta$$. The left hand limit of $$f(\theta)$$ as $$\theta \rightarrow \dfrac{\pi}{2}$$ is:
  • 1900
  • 2000
  • 2100
  • 2200
Consider $$A=\begin{bmatrix} \cos { \theta  }  & \sin { \theta  }  \\ -\sin { \theta  }  & \cos { \theta  }  \end{bmatrix}$$, then the value of $$\lim_{n \rightarrow \infty} \dfrac{A^{n}}{n}$$ (where $$\theta \in R$$) is equal to 
  • $$10$$
  • zero matrix
  • symmetric matrix
  • $$4$$
If $$\displaystyle\lim_ { x\rightarrow \lambda  } { \left( 2-\dfrac { \lambda  }{ x }  \right)  }^{ \lambda tan\left( \dfrac { \pi x }{ 2\lambda  }  \right)  }=\frac { 1 }{ e } ,$$ then $$\lambda $$ is equal to-
  • $$-\pi $$
  • $$\pi $$
  • $$\dfrac{\pi }{2}$$
  • $$-\dfrac{2}{\pi }$$
$$\displaystyle \lim_{x \rightarrow 0}\dfrac {1}{x\sqrt {x}}\left(a\ arc\ tan \dfrac {\sqrt {x}}{a}-b\ arc\ \tan \dfrac {\sqrt {x}}{b}\right)$$ has the value equal to
  • $$\dfrac {a-b}{3}$$
  • $$0$$
  • $$\dfrac {(a^{2}-b^{2})}{6a^{2}b^{2}}$$
  • $$\dfrac {a^{2}-b^{2}}{3a^{2}b^{2}}$$
$$\displaystyle \lim _{ x\rightarrow 2 }{ \frac { \sqrt [ 3 ]{ 60+{ x }^{ 2 } } -4 }{ \sin { \left( x-2 \right)  }  }  } $$
  • $$\dfrac {1}{4}$$
  • $$0$$
  • $$\dfrac {1}{12}$$
  • $$Does\ not\ exist$$
$$if\left( x \right)$$= greatest integer $$\le x$$, then $$\underset { x\longrightarrow 2 }{ lim }  { \left( -1 \right)  }^{ \left[ x \right]  } is  equal to-$$ is equal to-
  • 0
  • -1
  • +1
  • none of these
The value of $$\displaystyle \lim _{ x\rightarrow 0 }{ \csc^{ 4 }{ x } \int _{ 0 }^{ { x }^{ 2 } }{ \frac { ln\left( 1+4t \right)  }{ { t }^{ 2 }+1 }  } dt } $$ is 
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
The value of $$\displaystyle\lim_{n\rightarrow \infty}n(n\{ln (n)-ln (n+1)\}+1)$$ is?
  • e
  • $$\dfrac{1}{e}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{1}{4}$$
$$\displaystyle \lim_{x\rightarrow \infty}{x^{2}\sin\left(\log_{e}\sqrt{\cos\dfrac{\pi}{x}}\right)}$$
  • $$0$$
  • $$-\dfrac{\pi^{2}}{2}$$
  • $$-\dfrac{\pi^{2}}{4}$$
  • $$-\dfrac{\pi^{2}}{8}$$
$$ \underset { x\rightarrow \cfrac { \pi  }{ 2 }  }{ lim } \cfrac { cot \,  x-cos\, x }{ \left( \pi -{ 2x } \right)^ 3 } $$ equals
  • $$ \cfrac {1} {24} $$
  • $$ \cfrac {1} {16} $$
  • $$ \cfrac {1} {8} $$
  • $$ \cfrac {1} {4} $$
$$\underset { x\rightarrow \frac { \pi  }{ 2 }  }{ lim } \frac { (1-sinx)({ 8x }^{ 2 }-{ \pi  }^{ 3 })cosx }{ { (\pi -2x) }^{ 4 } } $$
  • $$-\cfrac { { \pi }^{ 2 } }{ 16 } $$
  • $$\cfrac { { 3\pi }^{ 2 } }{ 16 } $$
  • $$\cfrac { { \pi }^{ 2 } }{ 16 } \quad $$
  • $$-\cfrac { 3{ \pi }^{ 2 } }{ 16 } \quad $$
$$ \underset { x\rightarrow 0 }{ lim } \left[ { x }^{ 2 }cosec\quad \left( { x }^{ 2 } \right)^0 \right]  $$is equal to :
  • $$\cfrac { \pi} {180} $$
  • $$ \cfrac { \pi} {90} $$
  • $$ {0}  $$
  • $$\cfrac {90} { \pi } $$
The value of $$\lim_{x \rightarrow -1} \dfrac{\sqrt{\pi}-\sqrt{\cos^{-1}x}}{\sqrt{x+1}}$$ is given by 
  • $$\dfrac{1}{\sqrt{2}\sqrt{\pi}}$$
  • $$\dfrac{1}{\sqrt{\sqrt{2\pi}}}$$
  • $$1$$
  • $$0$$
If $$\lim_{x \rightarrow 0}\dfrac{a \sin x-bx+cx^{2}+x^{3}}{2x^{2} \log(1+x)-2x^{3}+x^{4}}$$ exists and is finite, then the value of $$a,b,c$$ are respectively 
  • $$0,6,6$$
  • $$6,0,6$$
  • $$6,6,0$$
  • $$0,0,6$$
$$ \underset { x\rightarrow a }{ lim } \cfrac { sin\quad x-sin\quad a }{ \sqrt [ 3 ]{ x } -\sqrt [ 3 ]{ a }  }  $$
  • $$ \sqrt [ 3 ]{ a\quad cos\quad a } $$
  • $$2\sqrt [ 3 ]{ a } $$
  • $$ { 3a }^{ { 2 /}{ 3 } }cos\quad a $$
  • $$ \sqrt [ 2 ]{ a\quad cos\quad a } $$
The value of $$\displaystyle \lim_{x\rightarrow 0}\dfrac {1-\cos^{3}x}{x\sin x\cos x}$$ is
  • $$\dfrac {2}{5}$$
  • $$\dfrac {3}{5}$$
  • $$\dfrac {3}{2}$$
  • $$\dfrac {3}{4}$$
Integrate:
 $$lim_{x\rightarrow 0}\dfrac{(1-\cos{2x})^{2}}{2x\tan{x}-x\tan{2x}}$$
  • $$2$$
  • $$\dfrac{-1}{2}$$
  • $$-2$$
  • $$\dfrac{1}{2}$$
$$\displaystyle \lim_{x\rightarrow 0^{+}}{(\csc x)^{1/\log x}}$$=
  • $$e$$
  • $$e^{-1}$$
  • $$e^{2}$$
  • $$1$$
The value of $$\lim_{x \rightarrow 0} \left(\dfrac{\tan x}{x}\right)^{1/x^{3}}$$ is-
  • $$0$$
  • $$\infty$$
  • $$e^{1/4}$$
  • $$Does\ not\ exist$$
The value of $$ \underset { x\rightarrow \frac { x }{ 2 }  }{ lim } \frac { log\sin { x }  }{ { \left( \frac { \pi  }{ 2 } -x \right)  }^{ 2 } }$$ is 
  • $$0$$
  • $$\frac{1}{2}$$
  • $$-\frac{1}{2}$$
  • $$-2$$
$$\lim_{n\rightarrow \infty}\dfrac{1}{n^{2}}\left[\sin^{3}\dfrac{\pi}{4n}+2\sin^{3}\dfrac{2\pi}{4n}+3\sin^{3}\dfrac{3\pi}{4n}+....+n\sin^{3}\dfrac{n\pi}{4n}\right]=$$
  • $$\dfrac{\sqrt{2}}{9\pi^{2}}\left(52-15\pi\right)$$
  • $$\dfrac{\sqrt{2}}{9\pi^{2}}\left(52+15\pi\right)$$
  • $$\dfrac{\sqrt{2}}{9\pi}\left(52-17\pi\right)$$
  • $$\dfrac{\sqrt{2}}{9\pi^{2}}\left(52+17\pi\right)$$
If $$ \alpha \quad and \beta $$ are the roots of the equation  $$ {ax}^{2}+bx+c=0 $$, then 
 $$ \underset { x\rightarrow \cfrac { \pi  }{ 2 }  }{ lim } \cfrac { tan\left[ \left( \alpha +\beta  \right) x \right]  }{ sin\left[ \left( \alpha \beta  \right) x \right]  }  $$ is equal to :
  • $$ \cfrac {c} {b} $$
  • $$ - \cfrac {b} {c} $$
  • $$ \cfrac {a} {b} $$
  • $$- \cfrac {a} {b} $$
$$\begin{matrix} lim \\ n\rightarrow \infty  \end{matrix}\int _{ 0 }^{ 1 }{ \frac { { nx }^{ n-1 } }{ 1+{ x }^{ 2 } }  } dx=$$
  • 0
  • 1
  • 2
  • $$\frac { 1 }{ 2 } $$
Arrange the following limits in the ascending order :
(1)  $$\lim _ { x \rightarrow \infty } \left( \dfrac { 1 + x } { 2 + x } \right) ^ { x + 2 }$$

(2)  $$\lim _ { x \rightarrow 0 } ( 1 + 2 x ) ^ { 3 / x }$$

(3)  $$\lim _ { \theta \rightarrow 0 } \dfrac { \sin \theta } { 2 \theta }$$

(4)  $$\lim _ { x \rightarrow 0 } \dfrac { \log _ { e } ( 1 + x ) } { x }$$
  • $$1,2,3,4$$
  • $$1,3,4,2$$
  • $$1,4,3,2$$
  • $$3,4,1,2$$
$$\mathop{\lim}\limits_{x \to 0} \left(\dfrac{3+x}{3-x}\right)^{\dfrac{x+1}{x}}$$ is equal to 
  • $$e^{2/3}$$
  • $$e^{1/3}$$
  • $$e^3$$
  • $$e^2$$
If $$\mathop {\lim }\limits_{x \to 0} \frac{{x\left( {1 + a\cos x} \right) - b\sin x}}{{{x^3}}} = 1,$$ then
  • $$a = \frac{5}{2}$$
  • $$b = \frac{{ - 5}}{2}$$
  • $$a + b = 4$$
  • $$a + b = -4$$
$$\lim_{x\rightarrow 0 }(\frac{p^{\frac{1}{x}}+q^{\frac{1}{x}}+r^{\frac{1}{x}}+s^{\frac{1}{x}}}{4})3x $$ where p,q,r,s$$> 0 $$ is equal to
  • $$pqrs$$
  • $$(pqrs)^{3}$$
  • $$(pqrs)\frac{3}{2}$$
  • $$(pqrs)\frac{3}{4}$$
$$\lim- {x\to 0}$$ $$\dfrac{1- cos(1 - cos4x)}{x^4}$$ is equal to : 
  • 4
  • 16
  • 32
  • None of these
The value of $$\displaystyle\lim_{x\to 0} |x|^{sinx}$$ equals 
  • $$0$$
  • $$-1$$
  • $$1$$
  • does not exist
If $$\displaystyle \lim _{ x\rightarrow 0 }{ \dfrac { \left( \sin { nx }  \right) \left[ (a-n)nx-tanx \right]  }{ { x }^{ 2 } }  } =0$$, then the value of $$a$$
  • $$\dfrac { 1 }{ n }$$
  • $$n-\dfrac { 1 }{ n } $$
  • $$n+\dfrac{1}{n}$$
  • $$None\ of\ these$$
$$\lim _ { x \rightarrow 0 } \frac { 1 - \cos x \cos 2 x \cos 3 x } { \sin ^ { 2 } 2 x } =$$
  • $$\frac { 3 } { 2 }$$
  • $$\frac { 5 } { 2 }$$
  • $$\frac { 7 } { 4 }$$
  • $$\frac { 9 } { 2 }$$
The value of $$lim_{x \to 0} (\dfrac{1}{x^2} - cotx)$$ equals 
  • $$1$$
  • $$0$$
  • $$\infty$$
  • Does not exist
$$ \displaystyle \lim _{ x-\infty  }{ sgn\left( \cot{\dfrac { { \pi x }^{ 2019 } }{ { x }^{ 2019 }+7 }}  \right)  }$$
  • Equals $$-1$$
  • Equals $$1$$
  • equals $$0$$
  • Does not exit
$$\displaystyle\lim_{x \to \pi/2} (sec x +tan x)$$ is equal to 
  • $$1$$
  • $$-1$$
  • $$\dfrac{1}{2}$$
  • $$0$$
$$\underset { x\rightarrow \pi/2 }{ lim } \left(\dfrac{cosec x-1}{cot^2x}\right)= $$
  • $$0$$
  • $$-\dfrac{1}{2}$$
  • $$\dfrac{1}{2}$$
  • $$1$$
$$\underset { x\rightarrow 0 }{ lim } \dfrac { x\tan { 2x } -2\tan { 2x }  }{ { \left( 1-cos2x \right)  } }$$ equals:
  • $$\dfrac{1}{4}$$
  • $$1$$
  • $$\dfrac{1}{2}$$
  • $$-\dfrac{1}{2}$$
$$\displaystyle\lim_{x\to \pi/2} \dfrac{sinx-(sinx)^{sin x}}{1-sin x + In sin x}$$ is equal to
  • $$4$$
  • $$2$$
  • $$1$$
  • none of these
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