Limits and Deri...

$$\underset { x\rightarrow 1 }{ lim } \frac { xtan(x-[x]) }{ x-1 } $$ is:

The value of $$\displaystyle \lim_{x \rightarrow 0} \left(\dfrac{\sin x}{x}\right)^{1/x^{2}}$$ is 

If z = z(x) and $$(2cosx)\frac { dz }{ dx } +(sinx)z=sinx$$, z(0) = 3, then $$z(\frac { \pi  }{ 2 } )$$ equals :

For the function, $$f(x) = (x - \frac{1}{x})^2$$, the first derivative with respect to x is 

The value of $$lim_{x\to 0} \dfrac{sin(\pi cos^2 x)}{x^2}$$ equals 

The value of $$lim_{\theta \to \dfrac{\pi}{2}} (sec \theta - tan \theta)$$ equals 

$$\lim- {x\to 0}$$ $$\dfrac{1- cos(1 - cos4x)}{x^4}$$ is equal to : 

The value of $$\displaystyle\lim_{x\to 0} |x|^{sinx}$$ equals 

If $$\displaystyle \lim _{ x\rightarrow 0 }{ \dfrac { \left( \sin { nx }  \right) \left[ (a-n)nx-tanx \right]  }{ { x }^{ 2 } }  } =0$$, then the value of $$a$$

Arrange the following limits in the ascending order :
(1)  $$\lim _ { x \rightarrow \infty } \left( \dfrac { 1 + x } { 2 + x } \right) ^ { x + 2 }$$

(2)  $$\lim _ { x \rightarrow 0 } ( 1 + 2 x ) ^ { 3 / x }$$

(3)  $$\lim _ { \theta \rightarrow 0 } \dfrac { \sin \theta } { 2 \theta }$$

(4)  $$\lim _ { x \rightarrow 0 } \dfrac { \log _ { e } ( 1 + x ) } { x }$$