From the uploaded image, we are solving the problem using the graph of the function $f(x)$. The questions involve determining specific function values and limits. Here's the breakdown:

Questions:

1. $f(1)$: The value of $f(x)$ at $x = 1$.
2. $\lim_{x \to 1} f(x)$: The limit of $f(x)$ as $x$ approaches 1.
3. $f(5)$: The value of $f(x)$ at $x = 5$.
4. $\lim_{x \to 5} f(x)$: The limit of $f(x)$ as $x$ approaches 5.

Step-by-Step Analysis:

(a) $f(1)$:

At $x = 1$, observe the graph. The point is clearly marked on the curve as $f(1) = 2$.

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

The left-hand limit and right-hand limit as $x \to 1$ converge at $y = 2$. Thus, $\lim_{x \to 1} f(x) = 2$.

(c) $f(5)$:

At $x = 5$, observe the graph. The graph is discontinuous at $x = 5$, and the solid point indicates $f(5) = 4$.

(d) $\lim_{x \to 5} f(x)$:

The left-hand limit approaches $y = 2$, but the right-hand limit approaches $y = 6$. Since the left-hand limit and right-hand limit do not match, the limit does not exist (DNE).

Final Answers:

(a) $f(1) = 2$
(b) $\lim_{x \to 1} f(x) = 2$
(c) $f(5) = 4$
(d) $\lim_{x \to 5} f(x) = \text{DNE}$

Let me know if you need detailed explanations for any step or have additional questions!

Related Questions:

1. What is the difference between a function value and a limit at a point?
2. How do you determine whether a limit exists from a graph?
3. Why does a jump discontinuity affect the existence of a limit?
4. Can a function have a defined value at a point where the limit does not exist?
5. How do left-hand and right-hand limits contribute to the overall limit?

Tip:

Always check both the left-hand and right-hand behavior of the graph when analyzing limits to ensure accuracy.