1. **Problem Statement:** We analyze graph 6 to answer three questions about the function $f$: (a) At what numbers $a$ does $\lim_{x \to a} f(x)$ not exist? (b) At what numbers $a$ is $f$ not continuous? (c) At what numbers $a$ does $\lim_{x \to a} f(x)$ exist but $f$ is not continuous at $a$? 2. **Understanding the graph:** - There is a vertical asymptote near $x=1$, so the limit does not exist there. - There is a hole on the $x$-axis near $x=3$ with a filled point above it, indicating a removable discontinuity. - There is a rising line segment to a filled point near $x=5$ and a separate horizontal ray starting from an open circle above that point, indicating a jump discontinuity. 3. **Step (a): Limits that do not exist** - At $x=1$, the vertical asymptote means $\lim_{x \to 1} f(x)$ does not exist. 4. **Step (b): Points where $f$ is not continuous** - At $x=1$, due to the vertical asymptote. - At $x=3$, because of the hole and the filled point above it. - At $x=5$, because the function jumps from the filled point to the open circle. 5. **Step (c): Limits exist but $f$ is not continuous** - At $x=3$, the limit exists (the function approaches the hole), but $f(3)$ is defined differently (filled point above), so discontinuous. - At $x=5$, the limit from the left exists and equals the filled point, but the function value is different (open circle above), so discontinuous. **Final answers:** - (a) $x=1$ - (b) $x=1, 3, 5$ - (c) $x=3, 5$