The graph of f is given. Use the graph to compute the quantities asked for. (If an answer does not exist, enter DNE.)

The x y-coordinate plane is given. The curve enters the window in the second quadrant, goes up and right, changes direction at the approximate point (−11.5, 8.1), goes down and right, crosses the x-axis at approximately x = −6, passes through the point (−4, −4), crosses the y-axis at approximately y = −8.7, changes direction at the approximate point (1.4, −9), goes up and right, passes through the point (4, −8), changes direction at the approximate point (9.1, −4.7), goes down and right, and exits the window in the fourth quadrant.
(a)
lim 
x→4 
f(x)
 
(b)
lim 
x→−4 
f(x)

Let's address the problem step by step using the description of the graph of the function \( f(x) \).

### (a) \( \lim_{x \to 4} f(x) \)

The limit \( \lim_{x \to 4} f(x) \) represents the value the function \( f(x) \) approaches as \( x \) gets closer and closer to 4 from both the left and right sides.

According to the description, the graph of the function passes through the point \( (4, -8) \), and no jumps or discontinuities are mentioned around \( x = 4 \). This implies that the function is continuous at \( x = 4 \). Therefore, the limit is equal to the function value at this point.

\[
\lim_{x \to 4} f(x) = -8
\]

### (b) \( \lim_{x \to -4} f(x) \)

Similarly, the limit \( \lim_{x \to -4} f(x) \) represents the value the function approaches as \( x \) gets closer to \(-4\). From the description, the graph passes through the point \( (-4, -4) \), and there are no jumps or discontinuities around \( x = -4 \), suggesting the function is also continuous at this point.

\[
\lim_{x \to -4} f(x) = -4
\]

### Summary:

- \( \lim_{x \to 4} f(x) = -8 \)
- \( \lim_{x \to -4} f(x) = -4 \)

Let me know if you'd like more details or have any questions! Here are five related questions that might help expand your understanding:

1. What does it mean for a function to be continuous at a point?
2. How can limits help in identifying discontinuities in a graph?
3. What is the formal definition of a limit?
4. How do left-hand and right-hand limits affect the existence of a limit?
5. What are the common types of discontinuities in a function?

**Tip:** When analyzing a limit from a graph, always check the behavior of the function from both sides of the point of interest. If the function approaches the same value from both directions, the limit exists and equals that value!

The graph of f is given. Use the graph to compute the quantities asked for. (If an answer does not exist, enter DNE.) The x y-coordinate plane is given. The curve enters the window in the second quadrant, goes up and right, changes direction at the approximate point (−11.5, 8.1), goes down and right, crosses the x-axis at approximately x = −6, passes through the point (−4, −4), crosses the y-axis at approximately y = −8.7, changes direction at the approximate point (1.4, −9), goes up and right, passes through the point (4, −8), changes direction at the approximate point (9.1, −4.7), goes down and right, and exits the window in the fourth quadrant. (a) lim x→4 f(x) (b) lim x→−4 f(x)

This problem focuses on calculating limits from a graph, particularly lim x→4 f(x) and lim x→−4 f(x). Learn how to analyze the behavior of a function using continuity and limits. Suitable for high school students in Grades 10-12.

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