Answer
$$-1$$
$$\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{x(x-1)}\right) \text { does not exist. }$$
Work Step by Step
\begin{align*}
\lim _{x \rightarrow 0}\left(\frac{1}{x}+\frac{1}{x(x-1)}\right)&=\lim _{x \rightarrow 0} \frac{(x-1)+1}{x(x-1)}\\
&=\lim _{x \rightarrow 0} \frac{1}{x-1}\\
&=-1
\end{align*}
and
\begin{aligned}
&\text { The limit } \\
& \lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{x(x-1)}\right) \\
& \text { does not exist. }\\
&-\text { As } x \rightarrow 0+, \text { we have } \frac{1}{x}-\frac{1}{x(x-1)}=\frac{(x-1)-1}{x(x-1)}=\frac{x-2}{x(x-1)} \rightarrow \infty\\
&-\text { As } x \rightarrow 0-\text {, we have } \frac{1}{x}-\frac{1}{x(x-1)}=\frac{(x-1)-1}{x(x-1)}=\frac{x-2}{x(x-1)} \rightarrow-\infty
\end{aligned}
