(1) $\lim _{x \rightarrow \infty} \mathrm{e}^{\frac{1}{x}}$;
(2) $\lim _{x \rightarrow 0} \ln \frac{\sin x}{x}$;
(3) $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{\frac{x}{2}}$;
(4) $\lim _{x \rightarrow 0}\left(1+3 \tan ^2 x\right)^{\cot ^2 x}$;
(6) $\lim _{x \rightarrow 0} \frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x \sqrt{1+\sin ^2 x}-x}$;