CBSE Questions for Class 11 Commerce Applied Mathematics Limits And Continuity     Quiz 13 - MCQExams.com

If $$ \displaystyle \lim _{x \rightarrow 0} \dfrac{x^{n}-\sin x^{n}}{x-\sin ^{n} x} $$ is non-zero finite, then $$ n $$ must be equal
  • 4
  • 1
  • 2
  • 3
If $$ L=\displaystyle \lim _{x \rightarrow 0} \dfrac{\sin x+a e^{x}+b e^{-x}+c \ln (1+x)}{x^{3}}=\infty $$

Equation $$ a x^{2}+b x+c=0 $$ has
  • real and equal roots
  • complex roots
  • unequal positive real roots
  • unequal roots
$$\displaystyle \lim _{x \rightarrow \infty} \dfrac{2+2 x+\sin 2 x}{(2 x+\sin 2 x) e^{\sin x}} $$ is equal to
  • 0
  • 1
  • -1
  • Does not exists
Value of $$L=\displaystyle\lim_{n\rightarrow \infty n}\dfrac{1}{4}\left[1.\left(\displaystyle\sum_{k=1}^{n}k\right)+2.\left(\sum_{k=1}^{n-1}k\right)+3.\left(\sum_{k=1}^{n-2}k\right)+...+n.1\right]$$ is
  • $$1/24$$
  • $$1/12$$
  • $$1/6$$
  • $$1/3$$
The value of $$\lim _{n \rightarrow \infty}\left[\tan \dfrac{\pi}{2 n} \tan \dfrac{2 \pi}{2 n} \cdots \tan \dfrac{n \pi}{2 n}\right]^{1 / n}$$ is
  • $$e$$
  • $$e^2$$
  • $$1$$
  • $$e^3$$
$$\displaystyle  \lim _{x \rightarrow \pi / 2} \dfrac{\sin (x \cos x)}{\cos (x \sin x)} $$ is equal to
  • 0
  • p/2
  • p
  • 2p
The value of $$ \displaystyle \lim _{x \rightarrow 0} \dfrac{\sqrt{\dfrac{1}{2}(1-\cos 2 x)}}{x} $$ is
  • 1
  • -1
  • 0
  • None of these
The value of $$\displaystyle \lim_{n\infty}\dfrac{1}{n^2}\left\{ sin^3\dfrac{\pi}{4n}+2sin^3\dfrac{2\pi}{4n} + ... + nsin^3\dfrac{n\pi}{4n}\right\}$$ is equal to 
  • $$\dfrac{\sqrt{2}}{9\pi^2}(52-15 \pi )$$
  • $$\dfrac{2}{9\pi^2}(52-15n)$$
  • $$\dfrac{1}{9\pi^2}(15n-15)$$
  • None of these
0:0:1


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