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