1. The problem asks whether as $x$ goes to negative infinity, $g(x)$ also goes to negative infinity. 2. From the graph description, the curve starts in the bottom left quadrant going downward, which corresponds to very large negative values of $x$ and corresponding negative values of $g(x)$. 3. Because as $x$ becomes very large negative, the function values also become very large negative (moving downwards), this means $\lim_{x \to -\infty} g(x) = -\infty$. 4. Therefore, the correct statement is: True.