Given the description of the graph, let's compute the requested limits and function values.

(a) $\lim_{{x \to 0^-}} f(x)$

This limit represents the behavior of the function as $x$ approaches 0 from the left. According to the description, the curve in the second quadrant goes up and right steeply and exits the window to the left of the y-axis, re-entering the first quadrant near the origin. There is a closed point at the origin.

Since the graph approaches the origin from the left (from negative values of $x$), and there is a closed point at (0, 0), we conclude that: $\lim_{{x \to 0^-}} f(x) = 0$

(b) $\lim_{{x \to 4^+}} f(x)$

This limit represents the behavior of the function as $x$ approaches 4 from the right. The graph crosses the x-axis at $(4, 0)$ with an open point, but there is no further description of the function for values of $x > 4$.

From the description, as $x$ approaches 4 from the right, it approaches an open point at $(4, 0)$, meaning the limit does exist and equals: $\lim_{{x \to 4^+}} f(x) = 0$

(c) $\lim_{{x \to 0}} f(x)$

This limit considers the behavior of the function as $x$ approaches 0 from both sides. From part (a), the left-hand limit is 0. Since the right-hand limit (from the right of $x = 0$) approaches the closed point at $(0, 0)$, both sides give the same value.

Thus, the two-sided limit is: $\lim_{{x \to 0}} f(x) = 0$

(d) $\lim_{{x \to 4}} f(x)$

This limit considers the behavior of the function as $x$ approaches 4 from both sides. From part (b), the limit from the right-hand side is 0. From the left-hand side, the curve passes through the point $(4, 4)$, indicating the left-hand limit is 4.

Since the left-hand and right-hand limits do not match, the two-sided limit does not exist: $\lim_{{x \to 4}} f(x) = \text{DNE}$

(e) $f(0)$

From the description, there is a closed point at the origin $(0, 0)$, meaning the function value at $x = 0$ is: $f(0) = 0$

(f) $f(4)$

There is a closed point at $(4, 4)$, indicating that the function value at $x = 4$ is: $f(4) = 4$

Summary of Results:

(a) $\lim_{{x \to 0^-}} f(x) = 0$
(b) $\lim_{{x \to 4^+}} f(x) = 0$
(c) $\lim_{{x \to 0}} f(x) = 0$
(d) $\lim_{{x \to 4}} f(x) = \text{DNE}$
(e) $f(0) = 0$
(f) $f(4) = 4$

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Would you like more details on these computations? Here are 5 related questions to expand this topic:

1. How do one-sided limits differ from two-sided limits?
2. What are the conditions for a limit to exist at a given point?
3. How do open and closed points on a graph affect the function's value?
4. How can discontinuities like jumps and holes be identified using limits?
5. What role do limits play in understanding the continuity of a function?

Tip: When evaluating limits graphically, always check the behavior of the function from both the left and right to identify possible discontinuities or mismatches.