Let's analyze the graph and the corresponding questions. The task involves determining the function values and limits based on the given graph.

1. $f(1)$:

   • From the graph, when $x = 1$, the value of the function $f(x)$ is 2 (solid black dot at $(1, 2)$).

2. $\lim_{x \to 0^-} f(x)$ (left-hand limit as $x \to 0$):

   • Approaching from the left, as $x$ approaches 0, the graph seems to be approaching the value 1 (solid dot at $(0, 1)$).

3. $\lim_{x \to 5^+} f(x)$ (right-hand limit as $x \to 5$):

   • As $x$ approaches 5 from the right, the graph approaches the value 3 (open circle at $(5, 3)$, so not defined exactly at 5, but approaches 3).

4. $\lim_{x \to 5} f(x)$ (two-sided limit):

   • Approaching from the left as $x \to 5^-$, the value of the function approaches 2.
   • Approaching from the right as $x \to 5^+$, the function approaches 3.
   • Since the left-hand and right-hand limits are not equal, the limit $\lim_{x \to 5} f(x)$ does not exist (DNE).

Summary of answers:

• $f(1) = 2$
• $\lim_{x \to 0^-} f(x) = 1$
• $\lim_{x \to 5^+} f(x) = 3$
• $\lim_{x \to 5} f(x) = \text{DNE}$

Would you like a detailed explanation of any specific step? Here are some related questions:

1. What is the difference between left-hand and right-hand limits?
2. Why does the two-sided limit not exist at $x = 5$?
3. How can we identify a discontinuity on a graph?
4. What is the significance of open and closed dots in determining limits?
5. How does a piecewise function influence the behavior of limits?

Tip: When checking limits graphically, always verify whether the left-hand and right-hand limits agree for the two-sided limit to exist.