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

The value of $$\underset { x\rightarrow 0 }{ lim } \frac { { 27 }^{ x }-{ 9 }^{ x }{ -3 }^{ x }+1 }{ \sqrt { 5 } -\sqrt { 4+cos\quad x }  } $$
  • $$\sqrt { 5 } { \left( log3 \right) }^{ 2 }$$
  • $$8\sqrt { 5 } log3$$
  • $$16\sqrt { 5 } log3$$
  • $$8\sqrt { 5 } { \left( log3 \right) }^{ 2 }$$
$$\lim _ { x \rightarrow 0 } \dfrac { \sin 4 x } { \tan 7 x } =$$
  • $$1$$
  • $$\dfrac { 4 } { 7 }$$
  • $$\dfrac { 7 } { 4 }$$
  • $$0$$
The value of $$\underset { x\rightarrow 0 }{ lim } \frac { { 27 }^{ x }-{ 9 }^{ x }-{ 3 }^{ x }+1 }{ \sqrt { 5 } -\sqrt { 4+cosx }  } $$ is 
  • $$\sqrt { 5 } { \left( log3 \right) }^{ 2 }$$
  • $$8\sqrt { 5 } log3$$
  • $$16\sqrt { 5 } log3$$
  • $$8\sqrt { 5 } { \left( log3 \right) }^{ 2 }$$
$$\displaystyle \lim _ { x \rightarrow 0 } \dfrac { | \cos ( \sin ( 3 x ) ) | - 1 } { x ^ { 2 } }$$   equals
  • $$\dfrac { - 9 } { 2 }$$
  • $$\dfrac { - 3 } { 2 }$$
  • $$\dfrac { 3 } { 2 }$$
  • $$\dfrac { 9 } { 2 }$$
$$\displaystyle\lim_{x\rightarrow \dfrac{\pi}{4}}\dfrac{\cos x-\sin x}{\left(\dfrac{\pi}{4}-x\right)(\cos x+\sin x)}=?$$
  • $$2$$
  • $$1$$
  • $$0$$
  • $$3$$
$$\displaystyle\lim_{x\rightarrow 0}\left\{\dfrac{log_e(1+x)}{x^2}+\dfrac{x-1}{x}\right\}$$ is equal to?
  • $$\dfrac{1}{2}$$
  • $$-\dfrac{1}{2}$$
  • $$1$$
  • None of these
Let [x] denote the greatest integer less than or equal to x. Then :
$$\underset { x\rightarrow 0 }{ lim } \dfrac { tan\left( \pi { sin }^{ 2 }x \right) +\left( \left| x \right| -sin \right) \left( x\left[ x \right]  \right) ^{ 2 } }{ { x }^{ 2 } } :$$
  • does not exist
  • equal $$\pi $$
  • equal 0
  • equal $$\pi $$ + 1
$$\underset { x\rightarrow \frac { \pi  }{ 2 }  }{ lim } \frac { cotx-cosx }{ { (\pi -2x) }^{ 3 } } $$ equals :
  • $$\frac { 1 }{ 8 } $$
  • $$\frac { 1 }{ 4 } $$
  • $$\frac { 1 }{ 24 } $$
  • $$\frac { 1 }{ 16 } $$
$$\displaystyle\lim _{ n\rightarrow \infty  }{ { n }^{ 2 }\left( { x }^{ \dfrac { 1 }{ n }  }-{ x }^{ \dfrac { 1 }{ n+1 }  } \right) ,x>0 } $$ is equal to 
  • $$0$$
  • $$e^{x}$$
  • $$log_ex$$
  • $$None\ of\ these$$
If $$\displaystyle \lim _{ x\rightarrow 0 }[1+ax+bx^{2}]^{(2/x)}=e^{3}$$, then 
  • $$a=3,\ b=0$$
  • $$a=\dfrac{3}{2},\ b\neq1$$
  • $$a=\dfrac{3}{2},\ b=R$$
  • $$a=2,\ b=3$$
If [.] deotes the greatest integer function then
$$\begin{matrix} lim \\ x\rightarrow \pi /2 \end{matrix}\left[ \frac { x-\frac { \pi  }{ 2 }  }{ cosx }  \right] $$ is equal to
  • $$1$$
  • $$-1$$
  • $$2$$
  • $$-2$$
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
$$\displaystyle x\xrightarrow { lim } a\left(\sin\frac{x-a}{2}\tan\frac{\pi x}{2a}  \right) $$
  • $$\displaystyle a/\pi$$
  • $$-\displaystyle a/\pi$$
  • $$\displaystyle \pi/a$$
  • $$-\displaystyle \pi/a$$
Value of $$\underset { x\rightarrow 0 }{ lim } \dfrac { \sqrt [ 3 ]{ 1+\tan { x }  } -\sqrt [ 3 ]{ 1-\tan { x }  }  }{ x } $$ is
  • $$\dfrac { 1 }{ 2 } $$
  • $$-\dfrac { 2 }{ 3 } $$
  • $$\dfrac { 1 }{ 3 } $$
  • $$0$$
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
The value of $$\displaystyle \lim_{x \rightarrow 0} (\sin x)^{\dfrac{1}{x}}+(1+x)^{(\sin x)})=0$$, where $$x > 0$$, is :
  • $$0$$
  • $$-1$$
  • $$1$$
  • $$2$$
The value of $$\begin{matrix} lim \\ x\rightarrow 0 \end{matrix}$$ $$[\frac{x}{sinx}],$$ where [.] represents the greatest inter function , is 
  • $$1$$
  • $$0$$
  • $$-1$$`
  • none of these
$$\underset { x\rightarrow 0 }{ lim } \left( \dfrac { 1+tanx }{ 1+sinx }  \right) ^{ cosecx }$$ is equal to
  • e
  • $$\dfrac { 1 }{ e } $$
  • 1
  • None of these
The value of $$\displaystyle \lim_{x\rightarrow 0}\left(\dfrac {1}{x^{2}}-\cot x\right)$$ equals
  • $$1$$
  • $$0$$
  • $$\infty$$
  • $$Does\ not\ exist$$
$$\underset { \theta \longrightarrow 0 }{ Lt } \dfrac { 3tan\theta -tan3\theta  }{ { 2\theta  }^{ 3 } } =$$
  • -4
  • 1/4`
  • 3/4
  • 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
$$\underset { x\rightarrow 0 }{ Lt } (1+sin\quad x)^{ cot\quad x }=$$ 
  • e
  • $$e^{ 2 }$$
  • $$e^{ 3 }$$
  • $$e^{ 4 }$$
For $$a > 1$$ then $$\displaystyle\lim_{x\rightarrow \infty}\dfrac{a^x-a^{-x}}{a^x+a^{-x}}=?$$
  • $$1$$
  • $$1/2$$
  • $$1/3$$
  • $$1/15$$
The value of $$\lim_{h\rightarrow 0}\left\{\dfrac {1}{h.(8+h)^{1/3}-\dfrac {1}{2h}}\right\}$$ equals
  • $$0$$
  • $$\dfrac {-4}{3}$$
  • $$\dfrac {-16}{3}$$
  • $$\dfrac {-1}{48}$$
$$\underset { x\rightarrow 0 }{ lim } { \left( cosecx \right)  }^{ \dfrac { 1 }{ logx }  }$$ is equal to
  • $$0$$
  • $$1$$
  • $$\dfrac { 1 }{ e } $$
  • None of these
$$\underset { h\rightarrow 0 }{ lim } \quad \frac { (2+h)cos(2+h)-2cos2 }{ h } =$$
  • cos2-2sin2
  • cos2+2sin2
  • sin2-2cos2
  • sin2+2cos2
$$\underset { x\rightarrow 0 }{ lim } \dfrac { 1 }{ { e }^{ 2 } } tan\left( \dfrac { \pi  }{ 4 } +x \right) ^{ 1/x }$$
  • 0
  • 1
  • -1
  • e
$$\lim_{x\rightarrow 1}\dfrac {1+\sin \pi\left(\dfrac {3x}{1+x^{2}}\right)}{1+\cos \pi x}$$ is equal to
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
$$lim_{x \to 0^-} \dfrac{x([x]+ \mid x \mid )sin[x]}{\mid x \mid}$$ is equal to
  • $$-sin1$$
  • 0
  • 1
  • $$sin1$$
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
  • $$\dfrac{1}{2}\tan\dfrac{\theta}{2}$$
  • $$\dfrac{1}{2}\cot\dfrac{\theta}{2}$$
  • $$-\dfrac{1}{2}\tan\dfrac{\theta}{2}$$`
  • $$-\dfrac{1}{2}\cot\dfrac{\theta}{2}$$
The value of $$\lim_{x\rightarrow 0}\left(\dfrac {e^{x}+e^{-x}-2}{x^{2}}\right)^{1/x^{2}}$$ equals
  • $$e^{1/2}$$
  • $$e^{1/4}$$
  • $$e^{1/3}$$
  • $$e^{1/12}$$
$$\lim _ { x \rightarrow 0 } \frac { \sqrt [ 3 ] { 1 + \sin x } - \sqrt [ 3 ] { 1 - \sin x } } { x } =$$
  • 0
  • 1
  • 3$$/ 2$$
  • 2$$/ 3$$
$$\displaystyle\underset{x\rightarrow \dfrac{\pi}{4}}{Lt}\dfrac{\sqrt{2}-\cos x-\sin x}{(4x-\pi)^2}=?$$
  • $$\dfrac{1}{16\sqrt{2}}$$
  • $$\dfrac{1}{32\sqrt{2}}$$
  • $$\dfrac{1}{16}$$
  • $$\dfrac{1}{8}$$
$$\lim _ { x \rightarrow 0 } \frac { \ln ( \sin 3 x ) } { \ln ( \sin x ) }$$ is equal to
  • 0
  • 1
  • 2
  • none of these
$$\underset { x\rightarrow \infty  }{ Lt } { 5 }^{ x }sin\left( \cfrac { a }{ { 5 }^{ x } }  \right) =$$
  • 0
  • 5
  • log5
  • None of these.
$$L\underset { x\rightarrow 0 }{ im } \frac { \sec { 4x-\sec { 2x }  }  }{ \sec { 3x-\sec { x }  }  }=$$
  • 3/2
  • 2/3
  • 1/3
  • 3/4
the value of $$\underset { x\rightarrow 0 }{ lim } \frac { sin\alpha X-sin\beta X }{ { e }^{ \alpha X }-{ e }^{ \beta X } } $$ equals
  • 0
  • 1
  • -1
  • $$\alpha -\beta $$
$$\mathop {\lim }\limits_{h \to 0} \frac{{\sin \left( {a + 3h} \right) - 3\sin \left( {a + 2h} \right) + 3\sin \left( {a + h} \right) - \sin a}}{{{h^3}}}$$ is equal to 
  • $$\cos a$$
  • $$ - \cos a$$
  • $$\sin a$$
  • $$\sin a\cos a$$
$$\lim _{ x\rightarrow -\infty  }{ \cfrac { x^{ 4 }\sin { \cfrac { 1 }{ x }  } +x^{ 2 } }{ 1+\left| x \right| ^{ 3 } } = } $$
  • 1
  • -1
  • 2
  • -3
$$L\underset { x\rightarrow \infty  }{ im } \left( \sin { \sqrt { x+1 } -\sin { \sqrt { x }  }  }  \right) =$$
  • 2
  • -2
  • 0
  • 1
$$ \lim _{x \rightarrow 0} \frac{1-\cos ^{3} x}{x \sin 2 x}= $$
  • 1$$ / 2 $$
  • 3$$ / 2 $$
  • 3$$ / 4 $$
  • 1$$ / 4 $$
The value of $$\underset { x\rightarrow 0 }{ lim } \left( \left( sinx \right) ^{ 1/x }+\left( \dfrac { 1 }{ x }  \right) ^{ sinx } \right) $$ equals
  • 0
  • 1
  • $$\infty $$
  • -1
$$\displaystyle {Lt}_{x\rightarrow 0}\dfrac{cos5x cos3x}{x(sin5x sin3x)}$$ = 
  • $$-4$$
  • $$4$$
  • $$-\dfrac{1}{4}$$
  • $$\dfrac{1}{4}$$
$$\mathop {\lim }\limits_{x \to \pi /2} \left[ {x\tan x - \left( {\frac{\pi }{2}} \right)\sec x} \right]$$ is equal to
  • 1
  • -1
  • 0
  • None of these
The value of $$\lim _{ x\rightarrow 0 }{ \left( { \left( \sin { x }  \right)  }^{ 1/x }+{ \left( \dfrac { 1 }{ x }  \right)  }^{ \sin { x }  } \right)  } $$
  • $$0$$
  • $$1$$
  • $$\infty$$
  • $$-1$$
The value of $$\theta ,\quad is$$
$$\underset { o\rightarrow o }{ lim } \quad \frac { { cos }^{ 2 }\left\{ 1-{ cos }^{ 2 }\quad \left( 1-{ cos }^{ 2 }\quad .....\left( { cos }^{ 2 }\left\{ 1-{ cos }^{ 2 }\theta  \right\}  \right)  \right)  \right\}  }{ sin\left( \frac { \pi (\sqrt { \theta +4 } -2 }{ \theta  }  \right)  } $$
  • $$\frac { \sqrt { 2 } }{ 4 } $$
  • $$\sqrt { 2 } $$
  • 1
  • 2
$$\underset { x\rightarrow \infty  }{ Lim } (sin\sqrt { x+1 } -sin\sqrt { x } )=$$
  • 2
  • -2
  • 0
  • 1
0:0:1


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