Limits and Cont...

$$\lim _ { x \rightarrow 0 } \frac { \sin x \sin \left( \frac { \pi } { 3 } + x \right) \sin \left( \frac { \pi } { 3 } - x \right) } { x } =$$

$$lim_{x\to \dfrac{\pi}{2}} tan^2x(\sqrt{2sin^2x + 3 sin x +4} - \sqrt{sin^2x + 6 sin x+2})$$ is equal to

$$\lim _ { x \rightarrow 0 } \frac { 1 - \cos x } { x \log ( 1 + x ) } =$$

$$P= \lim_{x \rightarrow o^{+}} (1+ \tan^{2} \sqrt{x})^{1/2x}=$$____

$$\displaystyle \lim _{ x\rightarrow 0 } \left(\dfrac{1^{1/x}+2^{1/x}+3^{1/x}+.....n^{1/x}}{n}\right)^{nx} ,\ n\ \epsilon \ N $$ is equal to

The values of $$\displaystyle\lim_{n\rightarrow \infty}\dfrac{\sqrt[4]{n^5+2}-\sqrt[3]{n^2+1}}{\sqrt[5]{n^4+2}-\sqrt[2]{n^3+1}}$$ is?

Let $$f(x)=\displaystyle\lim _{ n\rightarrow \infty  }{ \dfrac { { 2x }^{ 2n }\sin { \frac { 1 }{ x } +x }  }{ 1+{ x }^{ 2x } }  } $$ then find 

$$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { x\tan { 2x } -2x\tan { x }  }{ \left( 1-\cos { 2x }  \right) ^{ 2 } }  }$$ equal to 

Evaluate $$\displaystyle \lim _{ x\rightarrow 0 }{ \dfrac { \sin { \left[ \cos { x }  \right]  }  }{ 1+\left[ \cos { x }  \right]  }  } $$ ($$[.]$$ denotes the greatest integer function)

$$if\left( x \right) =\left[ x-3 \right] +\left[ x-4 \right] \quad for\quad x\epsilon R\quad then\quad \underset { x\rightarrow 3 }{ lim } f\left( x \right) =$$