https://file.aiquickdraw.com/mathbot/file/1728107804545iyz6lq6b.jpg

Let's analyze the graph and evaluate the given limits:

1. **Statement I: \(\lim_{{x \to -1^+}} f(x) = 1\)**  
   This is asking for the right-hand limit as \(x\) approaches \(-1\). Looking at the graph as \(x\) approaches \(-1\) from the right (i.e., from values greater than \(-1\)), the function approaches the value \(1\). Hence, **this statement is true**.

2. **Statement II: \(\lim_{{x \to -1}} f(x) = 2\)**  
   This asks for the overall limit as \(x\) approaches \(-1\) (from both sides). From the graph, as \(x\) approaches \(-1\) from the left, the function approaches \(2\), and from the right, the function approaches \(1\). Since the left and right limits do not match, the overall limit does not exist, meaning **this statement is false**.

3. **Statement III: \(\lim_{{x \to -5}} f(x) = 4\)**  
   As \(x\) approaches \(-5\), both from the left and right, the function clearly approaches the value \(4\), so **this statement is true**.

### Conclusion:
- Statement I is true.
- Statement II is false.
- Statement III is true.

Thus, the correct answer is **C. I and III only**.

Would you like further details on the limit analysis or have any additional questions?

Here are 5 related questions for further practice:
1. What is the left-hand limit \(\lim_{{x \to -1^-}} f(x)\)?
2. What is the value of \(f(-1)\)?
3. Does the function have any vertical asymptotes?
4. What is the behavior of the function as \(x\) approaches infinity?
5. What is the value of \(\lim_{{x \to 2^+}} f(x)\)?

**Tip**: For a limit to exist at a point, both the left-hand and right-hand limits must be equal.

Which of the following statements is/are true about the limit properties of the function f(x) as given in the graph?

Use this graph-based limit analysis to evaluate the right-hand, left-hand, and overall limits of the function f(x) at specific points like x = -1 and x = -5. Explore limit laws and continuity for students in Grades 11-12.

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