Answer
(a) $ \sqrt{2}$
(b) $-\sqrt{2}$
Work Step by Step
(a) \begin{aligned}
\lim _{x \rightarrow 1^{+}} \frac{\sqrt{2 x}(x-1)}{|x-1|} &=\lim _{x \rightarrow-1^{+}} \frac{\sqrt{2 x}(x-1)}{(x-1)}, \quad(|x-1 |=x-1 \text { for } x>1) \\
&=\lim _{x \rightarrow-1^{+}} \sqrt{2 x}\\
&=\sqrt{2}
\end{aligned}
(b)
\begin{aligned}
\lim _{x \rightarrow 1^{-}} \frac{\sqrt{2 x}(x-1)}{|x-1|} &=\lim _{x \rightarrow 1^{-}} \frac{\sqrt{2 x}(x-1)}{-(x-1)}, \quad(|x-1 |=-(x-1) \text { for } x<1) \\ &=\lim _{x \rightarrow 1^{-}}-\sqrt{2 x}\\
&=-\sqrt{2} \end{aligned}
