Let's go through the question.

You're asked to find the limits of the function $f(x)$ at $a = 0$ using the graph of $f(x)$ provided.

Step-by-Step Analysis:

1. Left-Hand Limit $\lim_{x \to 0^-} f(x)$:

   • As $x$ approaches 0 from the left (negative side), the graph shows $f(x)$ approaching a value of $2$.
   • This matches with the provided answer for $\lim_{x \to a^-} f(x) = 2$, which is marked correct.

2. Right-Hand Limit $\lim_{x \to 0^+} f(x)$:

   • As $x$ approaches 0 from the right (positive side), the function $f(x)$ goes down sharply. This suggests that $f(x)$ is approaching $-\infty$ (not $-2$, as entered in the answer).
   • Therefore, the correct answer for $\lim_{x \to 0^+} f(x)$ should be $-\infty$.

3. Two-Sided Limit $\lim_{x \to 0} f(x)$:

   • For a two-sided limit $\lim_{x \to 0} f(x)$ to exist, both the left-hand and right-hand limits must be equal.
   • Here, since $\lim_{x \to 0^-} f(x) = 2$ and $\lim_{x \to 0^+} f(x) = -\infty$, the two-sided limit does not exist (DNE).

Correct Answers:

• $\lim_{x \to 0^-} f(x) = 2$
• $\lim_{x \to 0^+} f(x) = -\infty$
• $\lim_{x \to 0} f(x) = \text{DNE}$

Would you like more details or have questions about these calculations?

Related Questions to Explore:

1. What is the difference between left-hand and right-hand limits?
2. Why do both left-hand and right-hand limits need to be equal for a two-sided limit to exist?
3. How can a limit approach infinity or negative infinity?
4. How do we interpret limits that result in $\pm \infty$ in terms of function behavior?
5. How can you use graphical information to estimate limits?

Tip:

When examining a graph for limits, always check both the left and right sides separately before assuming the two-sided limit exists.