If $f(x)= \begin{cases}\frac{8^x-4^x-2^x+1^x}{x^2}, & \text { if } x>0 \\ \mathrm{e}^x \sin x+\mathrm{i} x+\lambda \log 4, & \text { if } x \leqslant 0, \mathrm{i} \in \mathbb{R}\end{cases}$ continuous at $x=0$, then the value of $500 \mathrm{e}^\lambda$ is

$$ \mathop {\lim }\limits_{n \to \infty }\left[\frac{1}{1-n^4}+\frac{8}{1-n^4}+\ldots \ldots \ldots \ldots .+\frac{n^3}{1-n^4}\right]= $$

If $f(\theta)=\cos \theta_1 \cdot \cos \theta_2 \cdot \cos \theta_3$ .............. $\cos \theta_n$, then $\tan \theta_1+\tan \theta_2+\tan \theta_3+$. ............ $+\tan \theta_{\mathrm{n}}=$

$$ \mathop {\lim }\limits_{x \to \infty } \frac{\mathrm{e}^{x^4}-1}{\mathrm{e}^{x^4}+1}= $$