The image shows a graph and four options that ask about the behavior of the limits of $f(x)$ at different points, as well as the continuity of the function. Based on the graph, I can interpret the function and the question.

Analysis of the graph:

• The function is piecewise and consists of multiple segments and a smooth curve.
• It shows breaks or discontinuities at specific points, particularly at $x = -2$, $x = 0$, and $x = 2$.
• At $x = -2$, the graph appears to have a jump discontinuity.
• At $x = 0$, there is likely a removable discontinuity or a sharp change in direction.
• At $x = 2$, the function exhibits continuous behavior but is not necessarily smooth.

Question interpretation:

The problem asks to evaluate which of the following statements is correct regarding the limits of the function at specific points. The statements are:

1. $\lim_{x \to -2} f(x) = 1$
2. $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x)$
3. $\lim_{x \to 0} f(x) = 0$
4. $\lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x)$

Would you like to explore the details behind these limit evaluations, or do you need assistance with something specific from the question?

Here are some additional questions related to the topic of limits and continuity:

1. What are the conditions for a limit to exist at a point?
2. How does a jump discontinuity affect the limit of a function at that point?
3. Can a function be continuous but not differentiable? Provide an example.
4. What is the difference between one-sided limits and two-sided limits?
5. How do removable discontinuities differ from non-removable discontinuities?

Tip:

When evaluating limits graphically, always check the left-hand and right-hand behavior of the function at the point of interest.