Limits and Deri...

For $$x>y$$, $$\displaystyle\lim_{x\rightarrow 0}{\left[\left(\sin{x}\right)^{1/x}+\left(\cfrac{1}{x}\right)^{\sin{x}}\right]}$$ is :

If $$f : R \rightarrow (0, \infty)$$ is an increasing function and if $$\displaystyle\lim_{x \rightarrow 2018} \dfrac{f(3x)}{f(x)} = 1$$, then $$\displaystyle\lim_{x \rightarrow 2018} \dfrac{f(2x)}{f(x)}$$ is equal to 

If $$f$$ is differentiable at $$x = 1$$ and $$\underset{h \rightarrow 0}{\lim} \dfrac{1}{h} f (1 + h) = 5, f'(1) = $$

If $$y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+}}}.....\infty$$ then $$\dfrac{dy}{dx}=?$$

If $$y=(\tan x)^{\cot x}$$ then $$\dfrac{dy}{dx}=?$$

If $$x=a(\cos\theta+\theta\sin\theta)$$ and $$y=a(\sin\theta-\theta\cos\theta)$$ then $$\dfrac{dy}{dx}=?$$

If $$x=a\sec\theta, y=b\tan\theta$$ then $$\dfrac{dy}{dx}=$$

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

If $$y=\sqrt{\dfrac{1+\sin x}{1-\sin x}}$$ then $$\dfrac{dy}{dx}=?$$

The value of $$ \displaystyle \lim _{x \rightarrow \pi} \dfrac{1+\cos ^{3} x}{\sin ^{2} x} $$ is