Answer
Divergent
Work Step by Step
\[\begin{align}
& f\left( x \right)=\frac{1}{x-1},\text{ }\left( -\infty ,0 \right] \\
& \text{The area under the curve is given by } \\
& A=\int_{-\infty }^{0}{\frac{1}{x-1}}dx \\
& \text{By the definition of improper integrals} \\
& A=\underset{a\to -\infty }{\mathop{\lim }}\,\int_{a}^{0}{\frac{1}{x-1}}dx \\
& A=\underset{a\to -\infty }{\mathop{\lim }}\,\left[ \ln \left| x-1 \right| \right]_{a}^{0} \\
& A=\underset{a\to -\infty }{\mathop{\lim }}\,\left[ \ln \left| 0-1 \right|-\ln \left| a-1 \right| \right] \\
& A=\underset{a\to -\infty }{\mathop{\lim }}\,\left[ -\ln \left| a-1 \right| \right] \\
& \text{Evaluate when }a\to -\infty \\
& A=-\ln \left| -\infty -1 \right| \\
& A=\infty \\
& \text{The improper integral diverges.} \\
& \text{Divergent} \\
\end{align}\]
