6.54. $\lim _{x \rightarrow \pi / 2}(x-\pi / 2) \cdot \tan x$.
Solution.

$$
\begin{aligned}
& \lim _{x \rightarrow \pi / 2}(x-\pi / 2) \cdot \tan x=(0 \cdot \infty)=\lim _{x \rightarrow \pi / 2} \frac{x-\pi / 2}{\cot x}= \\
& =(0 / 0)=\lim _{x \rightarrow \pi / 2} \frac{(x-\pi / 2)'}{(\cot x)'}=\lim _{x \rightarrow \pi / 2} \frac{1}{-\frac{1}{\sin ^{2} x}}=-1
\end{aligned}
$$