Limits and Deri...

The value of $$\underset{x \rightarrow 0}{lim} \cos \,ec^4 \,x \displaystyle \overset{x^2}{\underset{0}{\int}} \dfrac{In(1 + 4t)}{1 + t^2}dt$$ is

If $$g(x)=\frac { x }{ \left[ x \right]  } for\quad x>2\quad then\quad \underset { x\rightarrow { 2 }^{ + } }{ Lim } \frac { g\left( x \right) -g\left( 2 \right)  }{ x-2 } $$

$$\lim_{x\rightarrow 0}\frac{ln(sin 3x)}{ln(sin x)}$$ is equal to

If $$\displaystyle \lim_{x\rightarrow 0}\dfrac {ae^{-x}-b\cos x-\dfrac {1}{2}cx}{x\cos x}=2$$ then the value of $$a+b+c$$ is-

$$\underset{x \rightarrow 2}{lim} \dfrac{\sqrt[3]{60 + x^2} - 4}{\sin (x - 2)}$$ equals 

$$\underset{x \rightarrow n/2}{lim} \dfrac{\cot x - \cos x}{(\pi - 2x)^3}$$ equals

$$\lim_{x\rightarrow 0}\dfrac{2\left(\sqrt{3}\sin\left(\dfrac{\pi}{6}+x\right)-\cos\left(\dfrac{\pi}{6}+x\right)\right)}{x\sqrt{3}\left(\sqrt{3}\cos x-\sin x\right)}$$

$$\lim_{x\rightarrow \pi/4}\dfrac{2\sqrt{2}-\left(\cos x+\sin x\right)^{2}}{1-\sin 2x}$$ is equal to 

$$\underset { x\rightarrow \pi /4 }{ Lim } \dfrac { 2\sqrt { 2 } \left( cosx+sinx \right) ^{ 3 } }{ 1-sin2x } =2$$ is equal to

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