Problem: Identify a number $a$ for each description (a)–(d) and for the final shown graph, and explain why. 1. (a) Statement: $\lim_{x\to a} g(x)$ does not exist but $g(a)$ is defined. Explanation: A jump discontinuity with left-hand limit $\lim_{x\to a^-} g(x)=1$ and right-hand limit $\lim_{x\to a^+} g(x)=3$ while the function value is $g(a)=2$ makes the two-sided limit fail to exist even though the point is defined. Reasoning: Because $1\neq3$ we conclude $\lim_{x\to a} g(x)$ does not exist while $g(a)$ exists. Choice and answer: $a=2$ since the example places the filled point at $(a,2)$. 2. (b) Statement: $\lim_{x\to a} g(x)$ exists but $g(a)$ is not defined. Explanation: This is a removable discontinuity (a hole) where both one-sided limits equal the same number, here $2$, but the function value at that $a$ is missing. Reasoning: If $\lim_{x\to a^-} g(x)=\lim_{x\to a^+} g(x)=2$ and the point $(a,2)$ is a hole, then the two-sided limit exists while $g(a)$ is undefined. Choice and answer: choose $a=5$ (top-right location) so the hole is at $(5,2)$ and $\lim_{x\to5} g(x)=2$ but $g(5)$ is undefined. 3. (c) Statement: $\lim_{x\to a^-} g(x)$ and $\lim_{x\to a^+} g(x)$ both exist but $\lim_{x\to a} g(x)$ does not exist. Explanation: This happens when the left and right one-sided limits exist but are unequal; here the left-hand limit equals the smaller value $1$ and the right-hand limit equals the larger value $3$. Reasoning: Since $\lim_{x\to a^-} g(x)=1$ and $\lim_{x\to a^+} g(x)=3$ with $1\neq3$, the two-sided limit $\lim_{x\to a} g(x)$ does not exist. Choice and answer: $a=1$ illustrates the top-left example, with smaller value $1$ and larger value $3$ for the one-sided limits. 4. (d) Statement: $\lim_{x\to a^+} g(x)=g(a)$ but $\lim_{x\to a^-} g(x)\neq g(a)$. Explanation: This describes a point where the right-hand limit equals the function value but the left-hand limit differs; for example $\lim_{x\to a^-} g(x)=0$, $\lim_{x\to a^+} g(x)=g(a)=2$, and there is a filled dot at $(a,2)$. Reasoning: Because the left and right one-sided limits are not equal, the overall two-sided limit may fail to equal the function value from the left, but the right-hand approach matches the point. Choice and answer: $a=4$ (bottom-right location) with $\lim_{x\to4^-} g(x)=0$, $\lim_{x\to4^+} g(x)=g(4)=2$ and the filled point $(4,2)$. 5. Final graph question: For the red piecewise curve described, find a number $a$ that satisfies one of the descriptions above. Observation: The graph has an open circle near $(3.5,2)$, a filled point at $(4,3)$, an open local maximum near $(2,3)$, and an open endpoint near $(6,0)$. Choice and explanation: The open circle at $(3.5,2)$ is a removable hole where both one-sided limits equal $2$ but the function is undefined there, matching description (b). Answer: $a=3.5$ since $\lim_{x\to3.5} g(x)=2$ but $g(3.5)$ is undefined.