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?

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!