1. The problem asks for the limit of the function $f(x)$ as $x$ approaches 2, i.e., $\lim_{x \to 2} f(x)$.\n\n2. The limit $\lim_{x \to a} f(x)$ exists if and only if the left-hand limit $\lim_{x \to a^-} f(x)$ and the right-hand limit $\lim_{x \to a^+} f(x)$ both exist and are equal.\n\n3. From the graph, as $x$ approaches 2 from the left, $f(x)$ approaches 2 (the horizontal segment at $y=2$ with an open circle at $(2,2)$). So, $\lim_{x \to 2^-} f(x) = 2$.\n\n4. As $x$ approaches 2 from the right, $f(x)$ approaches 4 (the horizontal segment at $y=4$ with an open circle at $(2,4)$). So, $\lim_{x \to 2^+} f(x) = 4$.\n\n5. Since $\lim_{x \to 2^-} f(x) = 2$ and $\lim_{x \to 2^+} f(x) = 4$ are not equal, the two-sided limit $\lim_{x \to 2} f(x)$ does not exist.\n\n6. Note that the value of the function at $x=2$ is $f(2) = 3$ (solid dot), but the limit depends on the behavior of $f(x)$ near 2, not the value at 2 itself.\n\nFinal answer: $\lim_{x \to 2} f(x)$ does not exist because the left and right limits are not equal.