Limits and Deri...

If $$x = a\sin 2\theta (1 + \cos 2\theta), y = b\cos 2\theta (1 - \cos 2\theta)$$, then $$\dfrac {dy}{dx} =$$

Evaluate: $$\underset { x\rightarrow 0 }{ \lim } \dfrac { x\tan2x-2x\tan x }{ \left( 1-\cos2x \right) ^{ 2 } } $$ 

$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \cot x - \cos x } { ( \pi - 2 x ) ^ { 3 } }$$ equals:

The solution of the differential equation  $$\left( \dfrac { d y } { d x } \right) ^ { 2 } - 3 x \left( \dfrac { d y } { d x } \right) - 2 y = 8$$  is

Evaluate: $$\underset { x\rightarrow { 0 } }{ lim } \dfrac { x\tan 2x-2x \tan\ x\quad  }{ (1-\cos2x)^{ 2 } } $$

Let $$f(x)=\begin{cases} { x }^{ 2 }+k,\quad \quad  when\quad x\ge 0 \\ -{ x }^{ 2 }-k,\quad \quad when \quad x<0 \end{cases}$$. If the function $$f(x)$$ be continous at $$x=0$$, then $$k=$$

$$\dfrac{d}{dx}\left[\dfrac{tan x - cot x}{tan x + cot x}\right]=$$

Evaluate: $$\displaystyle\lim_{x\to 10}\dfrac{x^2-100}{x-10}$$

Solve:

The value of $$\displaystyle \lim _{ x\rightarrow 0 }{ f\left( x \right) } $$ where $$f(x)=\dfrac {\cos (\sin x)-\cos x}{x^{4}}$$, is