Limits and Deri...

Evaluate $$\displaystyle \lim_{n\rightarrow \infty} \dfrac{1+2+3+...+n}{n^2}$$

If $$y=x^2\sin \dfrac{1}{x}$$ then $$\dfrac{dy}{dx}=?$$

If $$(x+y)=\sin (x+y)$$ then $$\dfrac{dy}{dx}=?$$

If $$x=a\cos^{2}\theta, y=b\sin^{2}\theta$$ then $$\dfrac{dy}{dx}=?$$

If $$y=e^{\sin \sqrt x}$$ then $$\dfrac{dy}{dx}=?$$

If $$y=\cos^2 x^3$$ then $$\dfrac{dy}{dx}=?$$

lf $$ \displaystyle \lim _{ x\rightarrow 0 } \left(\displaystyle  \frac { \cos  4x+a\cos  2x+b }{ x^{ 4 } }  \right) $$ is finite then the value of $$a,b$$ respectively are

$$\displaystyle \lim_{x\rightarrow 1}(1-x)\tan(\displaystyle \frac{\pi x}{2})=$$

The value of $$\displaystyle \lim _{ \alpha \rightarrow \beta  }{ \left[ \frac { \sin ^{ 2 }{ \alpha  } -\sin ^{ 2 }{ \beta  }  }{ { \alpha  }^{ 2 }-{ \beta  }^{ 2 } }  \right]  } $$ is:

Evaluate: $$\displaystyle \lim_{x\rightarrow 0}\displaystyle \frac{\sec 4x-\sec 2x}{\sec 3x-\sec x}$$