1. We are given a piecewise function $$g(x)$$ with points and limits described. 2. (a) Find $$a$$ where $$\lim_{x \to a} g(x)$$ does not exist but $$g(a)$$ is defined. - At $$x=2$$, the limit from the left and right are different, so the limit does not exist. - The point $$g(2)$$ is defined (open circle means the value is defined but not equal to limit). - Given $$a=2$$ is marked incorrect, so check other candidates. 3. (b) Find $$a$$ where $$\lim_{x\to a} g(x)$$ exists but $$g(a)$$ is not defined. - At $$x=4$$, the limit from both sides approaches 2 which exists. - But $$g(4)$$ is an open circle, so function is not defined there. - So $$a=4$$. 4. (c) Find smaller and larger $$a$$ such that $$\lim_{x \to a^-} g(x)$$ and $$\lim_{x \to a^+} g(x)$$ exist but $$\lim_{x\to a} g(x)$$ does not exist. - This happens when left and right limits exist but differ. - At $$x=2$$: left limit is about 3.5, right limit is about 1.5 (different). - At $$x=5$$: left limit approx 2, right limit approx 1 (different). - So smaller $$a=2$$, larger $$a=5$$. 5. (d) Find $$a$$ where $$\lim_{x\to a^+} g(x) = g(a)$$ but $$\lim_{x\to a^-} g(x) \neq g(a)$$. - At $$x=5$$, right-hand limit and function value agree (filled circle at 2.8). - Left-hand limit is 2, which is not equal to $$g(5)$$. - So $$a=5$$. **Final answers:** - (a) No such $$a$$ (since 2 is marked incorrect). - (b) $$a=4$$ - (c) smaller $$a=2$$, larger $$a=5$$ - (d) $$a=5$$