Let's break this down step by step by analyzing the graph and answering the given questions.

Part 1: $f(1)$

• From the graph, observe what happens at $x = 1$.
• At $x = 1$, there is a hole in the graph, indicating that $f(1)$ is undefined (meaning it does not exist at $x = 1$).
• Therefore, $f(1) = \text{DNE}$ (Does Not Exist).

Part 2: $\lim_{x \to 1} f(x)$

• The limit as $x \to 1$ looks at the behavior of the function as $x$ approaches 1 from both sides (left and right).
• As $x$ approaches 1 from the left, the graph approaches a value of approximately 0.
• As $x$ approaches 1 from the right, the graph also approaches the value 0.
• Since both the left-hand and right-hand limits agree, the limit exists and equals 0.

Thus, we have the following answers:

1. $f(1) = \text{DNE}$
2. $\lim_{x \to 1} f(x) = 0$

Would you like more details on any part of this? Here are 5 related questions to further your understanding:

1. What is the significance of a hole in a graph at a point like $x = 1$?
2. How can you determine if a limit exists at a point where the function itself is undefined?
3. What is the difference between a function value and a limit at the same point?
4. How can you find the limit at other points on the graph?
5. What role does continuity play in determining function values and limits?

Tip: When evaluating limits, always check the behavior of the function from both sides of the point of interest (left-hand limit and right-hand limit).