2.175. $\frac{\sqrt{x-2 \sqrt{x-1}}}{\sqrt{x-1}-1}$.
Solution.

Domain of definition: $1 \leq x \neq 2$.

$$
\begin{aligned}
& \frac{\sqrt{x-2 \sqrt{x-1}}}{\sqrt{x-1}-1}=\frac{\sqrt{x-1-2 \sqrt{x-1}+1}}{\sqrt{x-1}-1}=\frac{\sqrt{(\sqrt{x-1})^{2}-2 \sqrt{x-1}+1}}{\sqrt{x-1}-1}= \\
& =\frac{\sqrt{(\sqrt{x-1}-1)^{2}}}{\sqrt{x-1}-1}=\frac{|\sqrt{x-1}-1|}{\sqrt{x-1}-1}= \\
& =\left\{\begin{array}{l}
-\frac{\sqrt{x-1}-1}{\sqrt{x-1}-1}=-1 \text { for } \sqrt{x-1}-1<0 \text { or } x<2 ; \\
\frac{\sqrt{x-1}-1}{\sqrt{x-1}-1}=1 \text { for } \sqrt{x-1}-1>0 \text { or } x>2 .
\end{array}\right.
\end{aligned}
$$

Answer: -1 for $x \in[1 ; 2) ; 1$ for $x \in(2 ; \infty)$.