1. **Problem statement:** Find the limit $$\lim_{x \to 3} f(x)$$ where $$f(x) = \begin{cases} x - 1, & x \leq 3 \\ 3x - 7, & x > 3 \end{cases}$$ 2. **Formula and rules:** For piecewise functions, the limit at a point exists if and only if the left-hand limit and right-hand limit at that point are equal. 3. **Calculate left-hand limit:** $$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (x - 1) = 3 - 1 = 2$$ 4. **Calculate right-hand limit:** $$\lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} (3x - 7) = 3(3) - 7 = 9 - 7 = 2$$ 5. **Compare limits:** Since both left and right limits equal 2, the limit exists and is 2. 6. **Final answer:** $$\lim_{x \to 3} f(x) = 2$$