The question asks whether the statement $\lim_{x \to 0} f(x)$ exists is true or false, and we have a graph of the function to analyze this limit.

Analysis of the Graph:

To determine if the limit $\lim_{x \to 0} f(x)$ exists, we need to check the left-hand limit ($\lim_{x \to 0^-} f(x)$) and the right-hand limit ($\lim_{x \to 0^+} f(x)$).

• Left-hand limit ($\lim_{x \to 0^-} f(x)$): From the left of $x = 0$, the graph shows that as $x$ approaches 0, the function approaches $f(x) = -2$.

• Right-hand limit ($\lim_{x \to 0^+} f(x)$): From the right of $x = 0$, the graph shows that as $x$ approaches 0, the function also approaches $f(x) = -2$.

Since both the left-hand and right-hand limits are equal to $-2$, the overall limit $\lim_{x \to 0} f(x)$ exists and is equal to $-2$.

Answer:

The statement $\lim_{x \to 0} f(x)$ exists is true.

Would you like further details or explanations on this? Here are some related questions:

1. What does it mean for a limit to exist at a certain point?
2. How do you determine limits graphically?
3. What are left-hand and right-hand limits?
4. How would the answer change if the limits from both sides were not equal?
5. How does the continuity of a function relate to the existence of a limit?

Tip: When finding limits graphically, always compare the values from both the left and right sides of the point of interest.