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.