CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 14 - MCQExams.com

$$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { x\tan { 2x } -2x\tan { x }  }{ \left( 1-\cos { 2x }  \right) ^{ 2 } }  }$$ equal to 
  • $$\dfrac{1}{4}$$
  • $$1$$
  • $$\dfrac{1}{2}$$
  • $$None of these$$
$$\displaystyle\lim_{x\rightarrow \dfrac{\pi}{4}}\dfrac{\cos x-\sin x}{\left(\dfrac{\pi}{4}-x\right)(\cos x+\sin x)}=?$$
  • $$2$$
  • $$1$$
  • $$0$$
  • $$3$$
If $$ \lim _{x \rightarrow 0}\left(\cos x+a^{3} \sin \left(b^{6} x\right)\right)^{\frac{1}{x}}=e^{512} $$ then value of $$ab^2$$ is equal to
  • $$-512$$
  • $$512$$
  • $$8$$
  • none of these
$$if\left( x \right) =\left[ x-3 \right] +\left[ x-4 \right] \quad for\quad x\epsilon R\quad then\quad \underset { x\rightarrow 3 }{ lim } f\left( x \right) =$$
  • -2
  • -1
  • 0
  • 2
The value of $$\int _{ 0 }^{ \left[ x \right]  }{ \left( x-\left[ x \right]  \right) dx\quad is\left( \left[ . \right] denotes\quad greatest\quad integer\quad function \right)  } $$
  • $$\left[ x \right] $$
  • $$2\left[ x \right] $$
  • $$\dfrac { \left[ x \right] }{ 2 } $$
  • $$3\left[ x \right] $$
$$\underset { x\rightarrow 0 }{ lim } \dfrac { 1-\cos { 2x }  }{ \cos { 2x } -\cos { 8x }  } $$ is equal to 
  • $$-1/15$$
  • $$1/10$$
  • $$1/15$$
  • $$15$$
The value of $$\underset { x\rightarrow 0 }{ lim } \frac { { 27 }^{ x }-{ 9 }^{ x }-{ 3 }^{ x }+1 }{ \sqrt { 5 } -\sqrt { 4+cosx }  } $$ is 
  • $$\sqrt { 5 } { \left( log3 \right) }^{ 2 }$$
  • $$8\sqrt { 5 } log3$$
  • $$16\sqrt { 5 } log3$$
  • $$8\sqrt { 5 } { \left( log3 \right) }^{ 2 }$$
$$\lim _{ x\rightarrow 0 }{ \cfrac { x.{ 10 }^{ x }-x }{ 1-cosx } = } $$
  • $$\log { 10 } $$
  • $$2\log { 10 } $$
  • $$3\log { 10 } $$
  • $$4\log { 10 } $$
$$\displaystyle \lim _ { x \rightarrow 0 } \dfrac { | \cos ( \sin ( 3 x ) ) | - 1 } { x ^ { 2 } }$$   equals
  • $$\dfrac { - 9 } { 2 }$$
  • $$\dfrac { - 3 } { 2 }$$
  • $$\dfrac { 3 } { 2 }$$
  • $$\dfrac { 9 } { 2 }$$
$$\displaystyle \lim _ { x \rightarrow 0 } \left( \left[ \dfrac { - 5 \sin x } { x } \right] + \left[ \dfrac { 6 \sin x } { x } \right] \right)$$  (where  $$[ .]$$  denotes greatest integer function) is equal to
  • $$0$$
  • $$-12$$
  • $$1$$
  • $$2$$
If $$\underset { x\rightarrow 0 }{ lim } \dfrac { { x }^{ 3 } }{ \sqrt { a+x } (bx-sinx) } =1,$$ a > 0, then a + b is equal to 
  • 36
  • 37
  • 38
  • 40
$$\underset { x\rightarrow 0 }{ lim } \frac { sin({ 6x }^{ 2 }) }{ Incos({ 2x }^{ 2 }-x) } =$$
  • 12
  • -12
  • 6
  • -6
$$\underset { x\rightarrow -\infty  }{ lim } \frac { ({ 3x }^{ 4 }+{ 2x }^{ 2 })sin(\frac { 1 }{ x } )+{ |x| }^{ 3 }+5 }{ { |x }|^{ 3 }+{ |x| }^{ 2 }+|x|+1 } =$$
  • 2
  • 1
  • -2
  • -3
$$\underset { x\rightarrow \frac { \pi  }{ 2 }  }{ lim } \frac { cotx-cosx }{ { (\pi -2x) }^{ 3 } } $$ equals :
  • $$\frac { 1 }{ 8 } $$
  • $$\frac { 1 }{ 4 } $$
  • $$\frac { 1 }{ 24 } $$
  • $$\frac { 1 }{ 16 } $$
Let $$f\left( x \right)=asin\left| x \right| +{ be }^{ \left| x \right|  }$$ is differentiable when
  • $$a=-b$$
  • $$a=b$$
  • $$a-0$$
  • $$b=0$$
The value of $$\displaystyle \lim_{x\rightarrow 0}\left(\dfrac {1}{x^{2}}-\cot x\right)$$ equals
  • $$1$$
  • $$0$$
  • $$\infty$$
  • $$Does\ not\ exist$$
$$\underset { \theta \longrightarrow 0 }{ Lt } \dfrac { 3tan\theta -tan3\theta  }{ { 2\theta  }^{ 3 } } =$$
  • 1/4`
  • 3/4
  • 4
  • -4
$$\underset { x\rightarrow 0 }{ Lt\quad  } \frac { sec\quad x-1 }{ { \left( sec\quad x+\quad 1 \right)  }^{ 2 } } =$$
  • 1/8
  • 11/4
  • 3.2
  • 2
If $$u=f(x^{2}), v=g(x^{3}),f(x)=sinx, g^{1}(x)=cosx$$ then find $$\frac{du}{dv}$$
  • $$1$$
  • $$\frac{2}{3}$$
  • $$\frac{2sin x^{2}}{3xcos x^{3}}$$
  • $$\frac{2x^{2}}{3x^{3}}$$
If $$\begin{matrix} lim\quad  \\ x\rightarrow 0 \end{matrix}\dfrac { x\left( 1+acosx \right) -bsinx }{ { x }^{ 3 } } =1$$ then value of a + b 
  • -4
  • -6
  • 1
  • None of these
Value of $$\underset { x\rightarrow 0 }{ lim } \dfrac { \sqrt [ 3 ]{ 1+\tan { x }  } -\sqrt [ 3 ]{ 1-\tan { x }  }  }{ x } $$ is
  • $$\dfrac { 1 }{ 2 } $$
  • $$-\dfrac { 2 }{ 3 } $$
  • $$\dfrac { 1 }{ 3 } $$
  • $$0$$
$$\lim _{ x\rightarrow 0 }{ \frac { \sqrt [ 3 ]{ 1+\sin { x }  } -\sqrt [ 3 ]{ 1-\sin { x }  }  }{ x }  } =$$
  • $$0$$
  • $$1$$
  • $$\frac { 2 }{ 3 } $$
  • $$\frac { 3 }{ 2 } $$
Value of $$\underset { x\rightarrow \dfrac { \pi  }{ 2 }  }{ lim } \tan { x } .\ell nsin{ x }$$ is 
  • 0
  • $$\dfrac { 1 }{ 2 } $$
  • $$\dfrac { 3 }{ 4 } $$
  • None of these
If $$cosy=xcos(a+y)and\quad \cfrac { dy }{ dx } =\cfrac { k }{ 1+{ x }^{ 2 }-2xcosa } $$ then find value of k?
  • sin a
  • cos a
  • 1
  • -sin a
$$\underset { x\rightarrow 0 }{ lim } \left( \dfrac { 1+tanx }{ 1+sinx }  \right) ^{ cosecx }$$ is equal to
  • e
  • $$\dfrac { 1 }{ e } $$
  • 1
  • None of these
The value of $$\displaystyle \lim _{ x\rightarrow \infty } (|x^{2}|+x)\log{(x\cot^{-1}{x})}$$ is :
  • $$\dfrac{1}{3}$$
  • $$-\dfrac{1}{3}$$
  • $$\dfrac{2}{3}$$
  • $$-\dfrac{2}{3}$$
$$\displaystyle \lim _{ x\rightarrow \dfrac { \pi  }{ 2 }  }{ \dfrac { \sin { x }  }{ \cos ^{ -1 }{ \left[ \dfrac { 1 }{ 4 } \left( 3\sin { x } -\sin { 3x }  \right)  \right]  }  }  } $$, where [.] denotes greatest integer function is :
  • $$\dfrac{2}{\pi}$$
  • $$1$$
  • $$\dfrac{4}{\pi}$$
  • $$does\ not\ exist$$
$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \left( \frac { 1 + \cos x } { 1 - \cos x } \right) ^ { \sec x } =$$
  • e
  • $$e^2$$
  • $$e^3$$
  • $$e/4$$
$$Im _{  }{ \left( \dfrac { 1 }{ 1-\cos { \theta  } +i\sin { \theta  }  }  \right)  } $$ is equal to
  • $$\dfrac{1}{2}\tan\dfrac{\theta}{2}$$
  • $$\dfrac{1}{2}\cot\dfrac{\theta}{2}$$
  • $$-\dfrac{1}{2}\tan\dfrac{\theta}{2}$$`
  • $$-\dfrac{1}{2}\cot\dfrac{\theta}{2}$$
$$L\underset { x\rightarrow 0 }{ im } \frac { \sec { 4x-\sec { 2x }  }  }{ \sec { 3x-\sec { x }  }  }=$$
  • 3/2
  • 2/3
  • 1/3
  • 3/4
0:0:1


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