CBSE Questions for Class 11 Engineering Maths Limits And Derivatives Quiz 8 - MCQExams.com

Find the value of limit $$\displaystyle \lim _{ x\rightarrow \frac { \pi  }{ 6 }  }{ \frac { 2\sin ^{ 2 }{ x } +\sin { x-1 }  }{ 2\sin ^{ 2 }{ x } -3\sin { x+1 }  } = } $$.
  • $$0$$
  • $$3$$
  • $$-3$$
  • $$1$$
If $$f(x+y)=2f(x).f(y)$$ for all $$x,y$$, where $$f'(0)=3$$ and $$f'(4)=2$$ then $$f'(4)=3$$ is equal to
  • 6
  • 12
  • 4
  • 3
If $$y=(x+\sqrt{x^{2}+a^{2}})^{n}$$ then $$\dfrac{dy}{dx}=$$
  • $$y$$
  • $$ny$$
  • $$\dfrac{ny}{\sqrt{x^{2}+a^{2}}}$$
  • $$\dfrac{y}{\sqrt{x^{2}+a^{2}}}$$
If $$f(x)$$ is the integral of $$\dfrac{2\sin{x}-\sin{2x}}{x^{3}},\ x\neq 0$$. Find $$\lim _{ x\rightarrow 0 }{ f^{ ' }\left( x \right)  } $$, where $$f^{ ' }\left( x \right) =\dfrac{df{(x)}}{dx}$$
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$\dfrac{1}{3}$$
  • $$2$$
$$\lim\limits_{x\rightarrow0}\dfrac{x\tan 2x-2x\tan x}{\left(1-\cos 2x\right)^{2}}=$$
  • $$2$$
  • $$\dfrac{1}{2}$$
  • $$-2$$
  • $$-\dfrac{1}{2}$$
$$\displaystyle\lim_{h\rightarrow 0}\dfrac{\sin\sqrt{x+h}-\sin\sqrt{x}}{h}=$$__________.
  • $$\cos\sqrt{x}$$
  • $$\dfrac{1}{2\sin \sqrt{x}}$$
  • $$\dfrac{\cos \sqrt{x}}{2\sqrt{x}}$$
  • $$\sin\sqrt{x}$$
If $$2f(\sin x)+\sqrt {2}f(\cos x)=\tan x,\ (x> 0)$$, then $$\displaystyle \lim _{ x\rightarrow 1 }{ \sqrt { 1-x } f\left( x \right) = }$$ 
  • $$\sqrt {2}$$
  • $$\dfrac {1}{\sqrt {2}}$$
  • $$-\sqrt {2}$$
  • $$-\dfrac {1}{\sqrt {2}}$$
$$\dfrac{d}{dx}\left[\dfrac{tan x - cot x}{tan x + cot x}\right]=$$
  • 2 sin2x
  • -2 sin2x
  • 2 cos 2x
  • -2cos 2x
If $$\underset { x\rightarrow { 0 } }{ lim } \dfrac { \left( ax+b \right) -\sqrt { 4+\sin x }  }{ \tan\quad x } =\dfrac { 27 }{ 4 } ~where ~a,b\in R$$ then the value of 
  • $$a = 2~ and~ b = 7$$
  • $$a = -2~and~ b = 7 $$
  • $$a = 7~ and~ b = 2 $$
  • $$a=7~ and ~b = -2$$
Evaluate: $$\underset { x\rightarrow 0 }{ \lim } \dfrac { x\tan2x-2x\tan x }{ \left( 1-\cos2x \right) ^{ 2 } } $$ 
  • $$\dfrac { 1 }{ 4 } $$
  • $$1$$
  • $$\dfrac { 1 }{ 2 } $$
  • $$-\dfrac { 1 }{ 2 } $$
$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \cot x - \cos x } { ( \pi - 2 x ) ^ { 3 } }$$ equals:
  • $$\frac { 1 } { 24 }$$
  • $$\frac { 1 } { 16 }$$
  • 0
  • $$\frac { 1 } { 4 }$$
The solution of the differential equation  $$\left( \dfrac { d y } { d x } \right) ^ { 2 } - 3 x \left( \dfrac { d y } { d x } \right) - 2 y = 8$$  is
  • $$y = 2 x ^ { 2 } + 4$$
  • $$y = 2 x ^ { 2 } - 4$$
  • $$y = 2 x + 4$$
  • $$y = 2 x - 4$$
$$\underset {x\rightarrow0}{ lim } \dfrac{x \tan 2 x - 2 x \tan x}{(1 - \cos 2 x)^2}$$ equals 
  • $$ \dfrac{1}{4}$$
  • 1
  • $$ \dfrac{1}{2}$$
  • $$-\dfrac{1}{2}$$
Let $$f(x)=\begin{cases} { x }^{ 2 }+k,\quad \quad  when\quad x\ge 0 \\ -{ x }^{ 2 }-k,\quad \quad when \quad x<0 \end{cases}$$. If the function $$f(x)$$ be continous at $$x=0$$, then $$k=$$
  • $$0$$
  • $$1$$
  • $$2$$
  • $$-2$$
The value of $$\displaystyle \lim _{ x\rightarrow 0 }{ f\left( x \right) } $$ where $$f(x)=\dfrac {\cos (\sin x)-\cos x}{x^{4}}$$, is
  • $$2$$
  • $$\dfrac {1}{6}$$
  • $$\dfrac {2}{3}$$
  • $$-\dfrac {1}{3}$$
The value of $$\displaystyle \lim _{ x\rightarrow a }{ \frac { \sqrt { x-b } -\sqrt { a-b }  }{ { x }^{ 2 }-{ a }^{ 2 } }  } \left( a>b \right) $$ is
  • $$\dfrac {1}{4a}$$
  • $$\dfrac {1}{a\sqrt {a-b}}$$
  • $$\dfrac {1}{2a\sqrt {a-b}}$$
  • $$\dfrac {1}{4a\sqrt {a-b}}$$
$$\lim _ { x \rightarrow 0 } \dfrac { ( 27 + x ) ^ { 1 / 3 } - 3 } { 9 - ( 27 + x ) ^ { 2 / 3 } }$$  equals :
  • $$- 1 / 6$$
  • $$1 / 6$$
  • $$1 / 3$$
  • $$- 1 / 3$$
A curve in the $$1^{st}$$ quadrant passes through $$(1,1)$$. Its drifferential equation is $$(y-xy^{2})dx+(x+x^{2}y^{2})dy=0$$. Hence the equation of the curve is 
  • $$y-\dfrac{1}{xy}=\ln y$$
  • $$y-\dfrac{1}{xy}=\ln x$$
  • $$y-xy=\ln y$$
  • $$y-xy=\ln x$$
If $$(cos x)^y = (sin y)^x$$, then $$\dfrac{dy}{dx}$$ =
  • $$\dfrac{log(siny) + y tan x}{ log (cos x) - x cot y}$$
  • $$\dfrac{log(siny) - y tan x}{ log (cos x) + cot y}$$
  • $$\dfrac{log(siny) }{ log (cos x) }$$
  • $$\dfrac{log(cos x) }{ log (sin y) }$$
$$\underset { x\rightarrow 1 }{ Lim } \left[ { \left[ \frac { 4 }{ { x }^{ 2 }-{ x }^{ -1 } } -\frac { { 1-3x+x }^{ 2 } }{ { 1-x }^{ 3 } }  \right]  }^{ -1 }+\frac { 3\left( { x }^{ 4 }-1 \right)  }{ { x }^{ 3 }-{ x }^{ -1 } }  \right] =$$
  • $$\frac { 1 }{ 3 } $$
  • 3
  • $$\frac { 1 }{ 2 } $$
  • $$\frac { 3 }{ 2 } $$
Let  $$U_{ { n } }=\dfrac { n! }{ (n+2)! } $$  where  $$n \in N .$$  If  $$S_{ { n } }=\sum _{ { n-1 } }^{ { n } } U_{ { n } }$$  then  $$\lim _ { n \rightarrow \infty } \mathrm { S } _ { n }$$  equals :
  • $$2$$
  • $$1$$
  • $$1/2$$
  • $$1/3$$
$$\displaystyle \underset { x\rightarrow 0 }{ lim } \ \ \frac { ({ 1-\cos2x) }^{ 2 } }{ 2x \tan x-x \tan2x } $$ is :
  • $$-2$$
  • $$\dfrac { -1 }{ 2 } $$
  • $$\dfrac { 1 }{ 2 } $$
  • $$2$$
The differential of $$f(x)=\sqrt{\dfrac{2-x}{2+x}}$$ at $$x=0$$ and $$\delta x=0.15$$ is
  • $$0.07$$
  • $$0.075$$
  • $$-0.075$$
  • $$0.15$$
The value of $$\displaystyle \lim_{x\rightarrow 0} \dfrac{(1-\cos 2x)\sin 5x}{x^{2}\sin 3x}$$ is
  • $$10/3$$
  • $$3/10$$
  • $$6/5$$
  • $$5/6$$
Let $$f(x)=\dfrac{ax+b}{x+1},lim_{x\rightarrow 0} f(x)=2$$ and $$lim_{x\rightarrow \infty} f(x)=1$$ then $$f(-2)=$$
  • $$1$$
  • $$2$$
  • $$-1$$
  • $$0$$
Evaluate the limit, $$\mathop {\lim }\limits_{x \to 0} \frac{{x({{(1 + x)}^{1/x}} - e)}}{{x({{(1 + {x^2})}^{1/{x^2}}} - e)}}$$
  • 0
  • 1
  • 2
  • DNE
Evaluate
$$\mathop {\lim }\limits_{x \to 0} \cfrac{{1 - \cos (1 - \cos 2x)}}{{{x^4}}}$$
  • $$4$$
  • $$2$$
  • $$1$$
  • $$\dfrac{1}{2}$$
$$\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } - x + 1 } - a x - b \right) = 0,$$   then the values of  $$a$$  and  $$b$$  are given by
  • $$a = - 1 , b = 1 / 2$$
  • $$a = 1 , b = 1 / 2$$
  • $$a = 1 , b = - 1 / 2$$
  • None of these
$$\lim _{ { x\rightarrow \pi /4 } } \dfrac { \cot ^{ { 3 } } x-\tan  x }{ \cos  (x+\pi /4) } $$  is
  • $$4$$
  • $$8 \sqrt { 2 }$$
  • $$8$$
  • $$4 \sqrt { 2 }$$
Let $$f(x)=\displaystyle\lim_{n\rightarrow \infty}\sum^{n-1}_{r=0}\dfrac{x}{(rx+1)\{(r+1)x+1\}}$$, then?
  • $$f(x)$$ is continuous but not differentiable at $$x=0$$
  • $$f(x)$$ is both continuous and differentiable at $$x=0$$
  • $$f(x)$$ is neither continuous nor differentiable at $$x=0$$
  • $$f(x)$$ is a periodic function
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