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!