1. Let's start by stating the problem: We want to construct a table to explore the limit of a function as it approaches a point where the limit is negative. 2. Consider the function $f(x) = \frac{1}{x}$ and examine the limit as $x$ approaches 0 from the positive and negative sides. 3. The limit formula is $\lim_{x \to a} f(x)$, which means we look at values of $f(x)$ as $x$ gets closer to $a$. 4. For $f(x) = \frac{1}{x}$, as $x \to 0^+$ (from the right), $f(x) \to +\infty$, and as $x \to 0^-$ (from the left), $f(x) \to -\infty$. 5. To illustrate a negative limit, let's consider $f(x) = -2x + 1$ and find $\lim_{x \to 3} f(x)$. 6. Construct a table of values approaching 3 from both sides: | $x$ | $f(x) = -2x + 1$ | |-----|-----------------| | 2.9 | $-2(2.9) + 1 = -5.8 + 1 = -4.8$ | | 2.99 | $-2(2.99) + 1 = -5.98 + 1 = -4.98$ | | 3.01 | $-2(3.01) + 1 = -6.02 + 1 = -5.02$ | | 3.1 | $-2(3.1) + 1 = -6.2 + 1 = -5.2$ | 7. As $x$ approaches 3, $f(x)$ approaches $-5$, which is a negative number. 8. Therefore, $\lim_{x \to 3} (-2x + 1) = -5$. This table helps visualize how the function values approach the negative limit as $x$ gets closer to 3.