https://file.aiquickdraw.com/mathbot/file/17261011026080qy2xtmw.jpg

Let's analyze the graph of \( f(x) \) based on the given image to answer the questions.

### (a) \( f(-3) \)
At \( x = -3 \), the graph shows a point at approximately \( y = 6 \).
- **Answer: 6**

### (b) \( f(2) \)
At \( x = 2 \), the function is undefined since there is no closed dot on the graph at \( x = 2 \).
- **Answer: DNE (Does Not Exist)**

### (c) \( \lim_{x \to -3} f(x) \)
The limit as \( x \to -3 \) from both sides approaches the value of the function at \( x = -3 \), which is 6.
- **Answer: 6**

### (d) \( \lim_{x \to -3^+} f(x) \)
As \( x \) approaches \( -3 \) from the right, the graph approaches the point \( y = 6 \).
- **Answer: 6**

### (e) \( \lim_{x \to -2} f(x) \)
As \( x \) approaches \( -2 \) from both sides, the function value approaches 2.
- **Answer: 2**

### (f) \( \lim_{x \to -2^+} f(x) \)
As \( x \) approaches \( -2 \) from the right, the graph still approaches 2.
- **Answer: 2**

### (g) \( \lim_{x \to -2^-} f(x) \)
As \( x \) approaches \( -2 \) from the left, the limit is also 2.
- **Answer: 2**

### (h) \( \lim_{x \to -2} f(x) \) (repeated)
This is the same as (e).
- **Answer: 2**

### (i) \( \lim_{x \to 2^-} f(x) \)
As \( x \) approaches 2 from the left, the function approaches negative infinity.
- **Answer: \( -\infty \)**

### (j) \( \lim_{x \to 2^+} f(x) \)
As \( x \) approaches 2 from the right, the function approaches positive infinity.
- **Answer: \( +\infty \)**

### (k) \( \lim_{x \to -5} f(x) \)
As \( x \) approaches \( -5 \), the function approaches negative infinity.
- **Answer: \( -\infty \)**

### (l) \( \lim_{x \to 5^+} f(x) \)
As \( x \) approaches 5 from the right, the graph behaves erratically (oscillates), so the limit does not exist.
- **Answer: DNE**

### (m) \( \lim_{x \to 5} f(x) \)
The function oscillates heavily near \( x = 5 \), indicating the limit does not exist.
- **Answer: DNE**

### (n) \( \lim_{x \to 7} f(x) \)
As \( x \) approaches \( 7 \), the function value approaches \( y = 2 \).
- **Answer: 2**

### (o) Vertical Asymptotes
Vertical asymptotes exist where the function tends to \( \infty \) or \( -\infty \). From the graph, there appears to be a vertical asymptote at \( x = 2 \).
- **Answer: \( x = 2 \)**

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Let me know if you'd like further clarification on any of these answers.

### Follow-up questions:
1. What are the general conditions for a limit to exist?
2. How does the behavior of the function near \( x = 5 \) lead to a "does not exist" limit?
3. How do we identify vertical asymptotes on a graph?
4. Why does the limit approach infinity at \( x = 2 \)?
5. What other types of discontinuities can a function have?

**Tip:** A limit exists only if the left-hand and right-hand limits are equal at a given point.

Given the graph of f(x), evaluate the following, if they exist. If a limit does not exist, write DNE, ∞, or -∞ as appropriate. No work needs to be shown here.

Learn how to evaluate the limits of a function using a graph, identify vertical asymptotes, and understand function behavior as x approaches specific points. This problem covers important calculus concepts like continuity, one-sided limits, and vertical asymptotes, suitable for high school students in Grades 11-12.

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