Limits and Deri...

$$\underset { x\rightarrow 0 }{ Lt } \cfrac {tanx-x}{x^2tanx}$$ equals:

$$\dfrac { d }{ dx } \left[ \left( \dfrac { { tan }^{ 2 }2x-{ tan }^{ 2 }x }{ 1-{ tan }^{ 2 }2x{ tan }^{ 2 }x }  \right) cot3x \right]$$

$$\displaystyle\lim_{x\rightarrow \infty}\left(\dfrac{x+1}{2x+1}\right)^{x^2}$$ equals?

$$\underset { x\rightarrow \pi/2 }{ lim } \left(\dfrac{cosec x-1}{cot^2x}\right)= $$

$$\underset{x \rightarrow \infty}{lim} \dfrac{2 \tan^{-1} x}{\pi}$$ equals $$e^L$$ then $$L$$ is equal to

$$\displaystyle \lim _{ x\rightarrow x/2 } \dfrac { \left[ 1-\tan { \left( \dfrac { x }{ 2 }  \right)  }  \right] \left[ 1-\sin { x }  \right]  }{ \left[ 1+\tan { \left( \dfrac { x }{ 2 }  \right)  }  \right] \left[ \pi -2x \right] ^{ 3 } } $$ is

$$\displaystyle \lim _{ \theta \rightarrow 0 }{ \frac { 4\theta \left( \tan { \theta -2\theta \tan { \theta  }  }  \right)  }{ { \left( 1-\cos { 2\theta  }  \right)  }^{ 2 } }  } $$ is

$$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { 3\sin { \left( { x }^{ 9} \right) -\sin { \left( { x }^{ 9 } \right)  }  }  }{ { x }^{ 3 } }  } =q$$

evaluate$$ \underset { x\rightarrow 0 }{ lim } \frac { x-\int _{ 0 }^{ x }{ { cost }^{ 2 }dt }  }{ { x }^{ 3 }-6x } $$

If $$k$$  is an integer such that $$\lim_{n \rightarrow \infty} \left[\left(\cos \dfrac{k\pi}{4}\right)^{2}-\left(\cos \dfrac{k\pi}{6}\right)^{2}\right]=0$$ then :