Self Studies

Limits and Cont...

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  • Question 1
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    Ltx0(cosecx1x)=?\displaystyle\underset{x\rightarrow 0}{Lt}\left(cosec x-\dfrac{1}{x}\right)=?

  • Question 2
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    If f(x)={12sinx π4x,ifxπ 4 a,ifx=π 4 f(x)=\begin{cases} \dfrac { 1-\sqrt { 2 } \sin { x }  }{ \pi -4x } ,\quad \quad ifx\neq \dfrac { \pi  }{ 4 }  \\ a\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ,\quad \quad ifx=\dfrac { \pi  }{ 4 }  \end{cases} is continous at x=π4x=\dfrac {\pi}{4} then a=a=

  • Question 3
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    IfAi=xaixai,i=1,2,3,.....nIf {A_i} = \frac{{x - {a_i}}}{{\left| {x - {a_i}} \right|}}, \,i = 1,2,3,.....n and a1<a2<a3....<an,then{a_1}< {a_2}< {a_3}....< {a_{n,}} \, then
    limxam(A1A2......An),1mn\mathop {\lim }\limits_{x \to {a_m}} \left( {{A_1}{A_2}......{A_n}} \right), 1 \le m \le n

  • Question 4
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    limxπ2cotxcosx(π2x)3\displaystyle \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\cot x - \cos x}}{{{{\left( {\frac{\pi }{2} - x} \right)}^3}}}

  • Question 5
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    limx0sin1xtan1x x3 \displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { \sin ^{ -1 }{ x } -\tan ^{ -1 }{ x }  }{ { x }^{ 3 } }  } is equal to 

  • Question 6
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    If limx(x2+x+1x+1axb)=4\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{x^2} + x + 1}}{{x + 1}} - ax - b} \right)\, = 4,then

  • Question 7
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    $$f(x)= x\sin\dfrac{1}{x} , \  for x\neq 0$$
           $$= 0,\  for x=0$$

    Then.

  • Question 8
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    limn(tanθ+12tanθ2+122tanθ22+...+12ntanθ2n)\displaystyle\lim_{n\rightarrow \infty}\left(\tan\theta +\dfrac{1}{2}\tan \dfrac{\theta}{2}+\dfrac{1}{2^2}\tan \dfrac{\theta}{2^2}+...+\dfrac{1}{2^n}\tan\dfrac{\theta}{2^n}\right) equals?

  • Question 9
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    Let p= limx0+(1+tan2x)12x\lim_{x\rightarrow 0+}(1+tan^{2}\sqrt{x})^{\frac{1}{2x}} then log p is equal to :

  • Question 10
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    f(x)={(3/x2)sin2x2 ifxM0 x2+2x+c13x2  if x0,x130 x=1/3f(x) = \left\{\begin{matrix}(3/x^{2})\sin 2x^{2} & if x M 0 \\\dfrac {x^{2} + 2x + c}{1 - 3x^{2}}  & if\ x \geq 0, x \neq \dfrac {1}{\sqrt {3}}\\ 0 & x = 1/ \sqrt {3}\end{matrix}\right. then in order that ff be continuous at x=0x = 0, the value of cc is

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