A graph of y= f(x) is shown and a c-value is given. Use the graph to find the following, whenever they exist. ( if the limit is infinite enter, ‘ infinity ‘ or negative infinity ‘, as appropriate. If the limit does not otherwise exist, enter DNE) c = -10 The xy-coordinate plane is given. A curve and a vertical dashed line are graphed. • a vertical dashed line crosses the x-axis at x = -10 • the curve with 2 parts enters the window almost horizontally above the x-axis, goes up and right becoming more steep, exits almost vertically just to the left of x = -10, reenters, goes down and right becoming less steep, crosses the y-axis at y= 3, and exits tge window almost horizontally above the x-axis (a) Lim f(x) x -> c^- (b) Lim f(x) x-> c^+ (c) Lim f(x) x-> c (d) f(c)

Let's break this down step by step using the details from the graph you've described for \( y = f(x) \), with \( c = -10 \).

### Observations:
1. There is a **vertical dashed line** at \( x = -10 \), suggesting a potential vertical asymptote at this point.
2. The curve:
   - Approaches \( x = -10 \) from the left, rising steeply upward.
   - Re-enters from the right of \( x = -10 \) by going downward, eventually crossing the y-axis at \( y = 3 \).

This suggests the curve behaves differently as \( x \) approaches \( c = -10 \) from the left and from the right.

### Now, let's answer the questions:

#### (a) \( \lim_{x \to -10^-} f(x) \)  
This asks for the limit as \( x \) approaches \( c = -10 \) from the left.

- From the left, the curve rises steeply as \( x \to -10^- \), heading upwards. This suggests the limit is approaching **positive infinity**.
  
Thus,  
\[ \lim_{x \to -10^-} f(x) = \infty \]

#### (b) \( \lim_{x \to -10^+} f(x) \)  
This asks for the limit as \( x \) approaches \( c = -10 \) from the right.

- From the right, the curve is going downward steeply as \( x \to -10^+ \), which indicates that the limit approaches **negative infinity**.

Thus,  
\[ \lim_{x \to -10^+} f(x) = -\infty \]

#### (c) \( \lim_{x \to -10} f(x) \)  
This asks for the general limit as \( x \) approaches \( -10 \) from both directions.

- Since the left-hand limit is \( \infty \) and the right-hand limit is \( -\infty \), the two one-sided limits do not agree. Therefore, the limit does **not exist (DNE)**.

Thus,  
\[ \lim_{x \to -10} f(x) = \text{DNE} \]

#### (d) \( f(-10) \)  
This asks for the actual value of the function at \( x = -10 \).

- From the description, it seems that the curve has a vertical asymptote at \( x = -10 \). When there is a vertical asymptote, the function is undefined at this point.

Thus,  
\[ f(-10) = \text{DNE} \]

### Final Answers:
(a) \( \infty \)  
(b) \( -\infty \)  
(c) DNE  
(d) DNE

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Would you like more details or have any other questions? Here are some related questions you could explore:

1. How do we determine if a limit exists at a specific point?
2. What is the significance of vertical asymptotes in a graph?
3. Can a function be continuous at a point if the limit does not exist?
4. What are the graphical indicators of an infinite limit?
5. How do vertical asymptotes affect the domain of a function?

**Tip:** When dealing with limits, always analyze the behavior from both the left and right sides for more clarity, especially near discontinuities.

A graph of y = f(x) is shown and a c-value is given. Use the graph to find the following, whenever they exist. (If the limit is infinite, enter 'infinity' or 'negative infinity,' as appropriate. If the limit does not otherwise exist, enter DNE) c = -10 The xy-coordinate plane is given. A curve and a vertical dashed line are graphed. • A vertical dashed line crosses the x-axis at x = -10. • The curve with 2 parts enters the window almost horizontally above the x-axis, goes up and right becoming more steep, exits almost vertically just to the left of x = -10, reenters, goes down and right becoming less steep, crosses the y-axis at y = 3, and exits the window almost horizontally above the x-axis (a) Lim f(x) x -> c^- (b) Lim f(x) x -> c^+ (c) Lim f(x) x -> c (d) f(c).

Analyze the limit behavior of f(x) as x approaches -10 from both sides using graph interpretation. This problem covers key calculus concepts including one-sided limits, vertical asymptotes, and function discontinuities. Ideal for high school students in Grades 10-12.

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