https://file.aiquickdraw.com/mathbot/file/1730763558985oaemaeqe.jpg

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?

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#### 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.

Use the graph of the function f to find the limits at the indicated value of a, if the limit exists. If an answer does not exist, enter DNE.

Learn how to find the left-hand, right-hand, and two-sided limits for a function with a discontinuity by analyzing its graph. This example includes an in-depth look at using visual information to understand the behavior of limits, including when limits approach infinity.

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