1. **State the problem:** Find the limit $$\lim_{x \to 3} f(x)$$ given the graph with specified points and shape. 2. **Analyze the graph at $x=3$:** The graph has an open circle at the point $(3,-2)$, which means $f(3)$ is not defined or is different from the value of the limit. 3. **Check values of $f(x)$ near $x=3$ from both sides:** - For $x$ approaching 3 from the left, the graph connects $(1,1)$ to $(3,-2)$ with a line that goes downward. - For $x$ approaching 3 from the right, the graph shows a horizontal line at $y=-2$ starting from $x=3$ extending rightwards. 4. **Calculate left-hand limit:** From the left, as $x \to 3^{-}$, the graph moves linearly from $(1,1)$ to $(3,-2)$. The slope $$m = \frac{-2 - 1}{3 - 1} = \frac{-3}{2} = -\frac{3}{2}$$. Equation of line segment: $$y - 1 = -\frac{3}{2}(x - 1) \Rightarrow y = -\frac{3}{2}x + \frac{3}{2} + 1 = -\frac{3}{2}x + \frac{5}{2}$$. Evaluate limit from left: $$\lim_{x \to 3^-} f(x) = -\frac{3}{2}(3) + \frac{5}{2} = -\frac{9}{2} + \frac{5}{2} = -2$$. 5. **Calculate right-hand limit:** From the right, as $x \to 3^{+}$, $f(x)$ is constant and equal to $-2$. Thus, $$\lim_{x \to 3^+} f(x) = -2$$. 6. **Conclusion:** Since left-hand and right-hand limits are equal, the limit exists: $$\lim_{x \to 3} f(x) = -2$$. Note: The function value $f(3)$ is not equal to the limit because the circle at $x=3$ is open, but this does not affect the limit. **Final answer:** $$\boxed{-2}$$