Let [ $x$ ] denote the greatest integer less than or equal to $x$ and $f(x)=2 x-[2 x]$. If $\mathop {\lim }\limits_{x \to {2^ - }} f(x)=l_1$ and $\mathop {\lim }\limits_{x \to {2^ + }} f(x)=l_2$, then $l_1+l_2=$

$$ \mathop {\lim }\limits_{x \to 0} \frac{\left(2^x-1\right)(1+\sin x)^{\frac{2}{\sin x}}}{\log (1+2 x)}= $$

Let $f(x)$ be a differentiable function such that $f(0)=0$ and $f^{\prime}(0)=20$. For $x \in\left(0, \frac{\pi}{2}\right]$, if

$A(x)=2 f(x) \operatorname{cosec} 4 x+4 f(x)\left(\cos ^2 x+1\right)-4 \cos ^2 x$, then $\mathop {\lim }\limits_{x \to 0} A(x)=$

If $x=\log _e\left(\cot \left(\frac{\pi}{4}+\theta\right)\right)$, then $\lim _{\theta \rightarrow 0} \frac{\theta}{(\sinh x)(\cosh x)}=$