The graph of f is given. Use the graph fo compute the quantifies asked for. ( if an answer does not exist, enter DNE)

The xy-coordinate plane is given.
• The curve with two parts enters the window in the second quadrant just above the x-axis, goes up and right getting more steep, passes through the point (-4, 4), exits the window in the second quadrant nearly vertically to the left of the y-axis, reenters in the first quadrant nearly vertically to the right of the y-axis, goes down and right getting less steep, crosses the x-axis at the open point (4, 0), and exits the window in the fourth quadrant nearly horizontally above
y = -4.
• There is a closed point at the origin.
• There is a closed point at (4, 4)

(a) Lim f(x)
x-> 0^-

(b) Lim f(x)
x-> 4^+

(c) Lim f(x)
x -> 0 

(d) Lim f(x)
x-> 4

(e) f(0)

(f) f(4)

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.

The graph of f is given. Use the graph to compute the quantities asked for. (if an answer does not exist, enter DNE). The xy-coordinate plane is given. The curve with two parts enters the window in the second quadrant just above the x-axis, goes up and right getting more steep, passes through the point (-4, 4), exits the window in the second quadrant nearly vertically to the left of the y-axis, reenters in the first quadrant nearly vertically to the right of the y-axis, goes down and right getting less steep, crosses the x-axis at the open point (4, 0), and exits the window in the fourth quadrant nearly horizontally above y = -4. There is a closed point at the origin. There is a closed point at (4, 4). (a) Lim f(x) x-> 0^- (b) Lim f(x) x-> 4^+ (c) Lim f(x) x -> 0 (d) Lim f(x) x-> 4 (e) f(0) (f) f(4)

Solve limit problems using a piecewise curve. This problem involves graphical analysis, one-sided limits, and discontinuities. Learn how to compute limits at key points such as 0 and 4, and understand function behavior from different sides of the graph. Suitable for students in Grades 11-12.

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