https://file.aiquickdraw.com/mathbot/file/1727897053980xyz1zgyk.jpg

The limit given in the image is:

\[
\lim_{{x \to -3}} \frac{|2x + 6|}{x + 3}
\]

### Step-by-Step Solution:

1. **Analyze the Absolute Value**: 
   The expression inside the absolute value is \(2x + 6\), which can be factored as:
   \[
   2x + 6 = 2(x + 3)
   \]
   Therefore:
   \[
   |2x + 6| = |2(x + 3)| = 2|x + 3|
   \]

2. **Simplify the Limit**:
   Substitute this into the original limit:
   \[
   \lim_{{x \to -3}} \frac{2|x + 3|}{x + 3}
   \]

3. **Behavior Near \(x = -3\)**:
   The behavior of \( |x + 3| \) depends on whether \(x\) approaches \(-3\) from the left or the right:
   - If \(x \to -3^-\) (approaching from the left), \(x + 3 < 0\), so \(|x + 3| = -(x + 3)\).
   - If \(x \to -3^+\) (approaching from the right), \(x + 3 > 0\), so \(|x + 3| = x + 3\).

4. **Left-Hand Limit** (\(x \to -3^-\)):
   In this case, \(|x + 3| = -(x + 3)\), so:
   \[
   \lim_{{x \to -3^-}} \frac{2|x + 3|}{x + 3} = \lim_{{x \to -3^-}} \frac{2(-(x + 3))}{x + 3} = \lim_{{x \to -3^-}} \frac{-2(x + 3)}{x + 3}
   \]
   For \(x \neq -3\), this simplifies to \(-2\). Thus, the left-hand limit is:
   \[
   \lim_{{x \to -3^-}} \frac{2|x + 3|}{x + 3} = -2
   \]

5. **Right-Hand Limit** (\(x \to -3^+\)):
   In this case, \(|x + 3| = x + 3\), so:
   \[
   \lim_{{x \to -3^+}} \frac{2|x + 3|}{x + 3} = \lim_{{x \to -3^+}} \frac{2(x + 3)}{x + 3} = \lim_{{x \to -3^+}} 2 = 2
   \]

6. **Conclusion**:
   Since the left-hand limit is \(-2\) and the right-hand limit is \(2\), the limit does **not exist** because the left and right limits are not equal.

### Final Answer:
\[
\lim_{{x \to -3}} \frac{|2x + 6|}{x + 3} \text{ does not exist.}
\]

Would you like more details or further clarification?

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Here are five related questions to expand your understanding:
1. What is the geometric interpretation of a limit that does not exist?
2. How does absolute value affect the behavior of functions near critical points?
3. How do left-hand and right-hand limits relate to the overall limit?
4. What are some strategies for solving limits involving piecewise functions?
5. How can you use graphical methods to determine if a limit exists?

**Tip**: Always check the left-hand and right-hand limits separately when dealing with absolute value expressions or discontinuities!

Find the limit: lim_{x→-3} |2x+6|/(x+3)

Learn how to solve the limit lim_{x→-3} |2x+6|/(x+3) step by step. This problem covers concepts like absolute values, piecewise functions, and how to compute limits from the left and right. Suitable for students in Grades 11-12.

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