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Let's break down the problem visible in your image.

### Problem Statement:

You are tasked with finding the limits of the function:

\[
f(x) = \frac{-3(x + 2)}{x^2 + 4x + 4}
\]

at three points:

1. \(\lim_{x \to -2} \frac{-3(x+2)}{x^2 + 4x + 4}\)
2. \(\lim_{x \to -2^-} \frac{-3(x+2)}{x^2 + 4x + 4}\) (left-hand limit)
3. \(\lim_{x \to -2^+} \frac{-3(x+2)}{x^2 + 4x + 4}\) (right-hand limit)

### Step-by-Step Solution:

1. **Factor the denominator**:
   
   The denominator is \(x^2 + 4x + 4\), which can be factored as:

   \[
   x^2 + 4x + 4 = (x + 2)^2
   \]

   Thus, the function becomes:

   \[
   f(x) = \frac{-3(x+2)}{(x+2)^2}
   \]

2. **Simplify the function**:

   After factoring, you can cancel out one of the \((x + 2)\) terms from the numerator and the denominator:

   \[
   f(x) = \frac{-3}{x + 2}
   \]

3. **Evaluate the limits**:

   - \(\lim_{x \to -2} \frac{-3}{x+2}\):

     As \(x\) approaches \(-2\), the denominator \(x + 2\) approaches 0. Since the numerator is \(-3\), this causes the limit to go to infinity. However, we must differentiate between the left-hand and right-hand limits because the behavior is different.

   - \(\lim_{x \to -2^-} \frac{-3}{x+2}\) (left-hand limit):

     As \(x\) approaches \(-2\) from the left (values less than \(-2\)), \(x + 2\) is negative and very small. This makes the function approach \(+\infty\) since a negative divided by a small negative number gives a positive large value.

   - \(\lim_{x \to -2^+} \frac{-3}{x+2}\) (right-hand limit):

     As \(x\) approaches \(-2\) from the right (values greater than \(-2\)), \(x + 2\) is positive and very small. This makes the function approach \(-\infty\) since a negative divided by a small positive number gives a large negative value.

### Final Results:

1. \(\lim_{x \to -2} \frac{-3}{x + 2}\) **does not exist** (DNE), because the left-hand and right-hand limits are not equal.
2. \(\lim_{x \to -2^-} \frac{-3}{x + 2} = +\infty\)
3. \(\lim_{x \to -2^+} \frac{-3}{x + 2} = -\infty\)

### Correction to the Image:
- The correct answers should be:
  1. \(DNE\)
  2. \(+\infty\)
  3. \(-\infty\)

Would you like further details or explanations on any of the steps?

Here are five related questions you could explore:
1. How do you differentiate between left-hand and right-hand limits?
2. What happens when the left-hand and right-hand limits are equal?
3. How do you factor quadratic expressions like \(x^2 + 4x + 4\)?
4. What is the formal definition of a vertical asymptote?
5. How do limits behave near infinity?

**Tip**: Always check for factorizations in the denominator to simplify limits involving polynomials!

Find the limits of the function -3(x + 2) / (x^2 + 4x + 4) as x approaches -2, from both sides and overall.

Solve the limit problem involving the rational function f(x) = -3(x + 2) / (x^2 + 4x + 4) and find its behavior as x approaches -2 from both sides. This problem explores the concept of vertical asymptotes and differentiates between left-hand and right-hand limits. Ideal for students in Grades 11-12 studying calculus.

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