https://file.aiquickdraw.com/mathbot/file/17237079142467fj9bqxa.jpg

Consider the graph of some function y=f(x). (shown in the picture).
**Evaluate** x→−6lim​f(x)**,** x→−2lim​f(x)**, and** x→4lim​f(x)**.**

To evaluate the limits \( \lim_{x \to -6} f(x) \), \( \lim_{x \to -2} f(x) \), and \( \lim_{x \to 4} f(x) \) from the provided graph, we'll inspect the behavior of the function \( f(x) \) as \( x \) approaches each of these values.

### 1. **Limit as \( x \) Approaches -6** (\( \lim_{x \to -6} f(x) \)):
   - Observe the graph as \( x \) approaches \(-6\).
   - The graph shows that as \( x \) approaches \(-6\) from both the left and right sides, \( f(x) \) appears to approach a specific value.
   - The function seems to approach a value of approximately \( y = 2 \).

   \[
   \lim_{x \to -6} f(x) = 2
   \]

### 2. **Limit as \( x \) Approaches -2** (\( \lim_{x \to -2} f(x) \)):
   - Now, observe the graph as \( x \) approaches \(-2\).
   - From the graph, it seems that as \( x \) approaches \(-2\) from both sides, the function \( f(x) \) approaches a specific value.
   - The function seems to approach a value of approximately \( y = 4 \).

   \[
   \lim_{x \to -2} f(x) = 4
   \]

### 3. **Limit as \( x \) Approaches 4** (\( \lim_{x \to 4} f(x) \)):
   - Finally, observe the graph as \( x \) approaches \( 4 \).
   - As \( x \) gets closer to \( 4 \) from both the left and right sides, the function \( f(x) \) seems to approach a specific value.
   - The function seems to approach a value of approximately \( y = 2 \).

   \[
   \lim_{x \to 4} f(x) = 2
   \]

### Summary of Limits:

\[
\lim_{x \to -6} f(x) = 2, \quad \lim_{x \to -2} f(x) = 4, \quad \lim_{x \to 4} f(x) = 2
\]

Would you like to go deeper into any specific limit evaluation, or have any other questions?

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### Follow-up Questions:
1. How do you determine if a limit exists at a certain point on a graph?
2. What does it mean for a function to have a left-hand or right-hand limit?
3. What if the graph had a jump discontinuity? How would that affect the limit?
4. How does the behavior of the graph near vertical asymptotes affect limits?
5. Can a limit exist if the function does not exist at that point?
6. What is the difference between a limit and the actual value of the function at a point?
7. How would you calculate limits for more complex functions?
8. What happens if the function approaches infinity as \( x \) approaches a certain value?

**Tip:** When analyzing limits graphically, it's crucial to observe the behavior of the function as it gets close to the point of interest from both directions (left and right). If both directions lead to the same value, the limit exists and equals that value.

Learn how to evaluate limits of a function using graphical analysis. Explore the behavior of functions as x approaches specific values from both the left and right sides. This problem involves identifying limit values from a graph, including detailed observations and interpretations. Suitable for advanced high school students.

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