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The given limit is:

\[
\lim_{{x \to 2^-}} \frac{x^2 + 4x + 4}{x - 2}
\]

We are asked to find the one-sided limit as \( x \) approaches 2 from the left, \( x \to 2^-\).

### Step-by-Step Solution:

1. **Simplify the numerator**: 
   The numerator is \( x^2 + 4x + 4 \). This can be factored as:
   \[
   x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2
   \]

   So the expression becomes:
   \[
   \frac{(x + 2)^2}{x - 2}
   \]

2. **Behavior at \( x = 2 \)**:
   If we substitute \( x = 2 \) directly into the expression, the denominator becomes zero, i.e., \( x - 2 = 0 \), which would result in an undefined form. Thus, we need to check the behavior as \( x \to 2^- \) (from the left).

3. **Analyzing the limit**:
   - As \( x \) approaches 2 from the left, \( (x + 2) \) approaches 4, and the squared term \( (x + 2)^2 \) approaches 16.
   - The denominator, \( x - 2 \), approaches 0 from the negative side, because when \( x \to 2^- \), \( x - 2 \) is a small negative number.

4. **Conclusion**:
   The numerator approaches 16, and the denominator approaches a small negative number. Therefore, the whole expression tends to negative infinity:

   \[
   \lim_{{x \to 2^-}} \frac{(x + 2)^2}{x - 2} = -\infty
   \]

Thus, the limit is \( L = -\infty \).

Let me know if you'd like further details or clarifications. Here are some related questions to explore:

1. What happens if we take the two-sided limit of the same expression?
2. How does the limit change if we approach \( x = 2^+ \) instead of \( x = 2^- \)?
3. How can you apply L'Hopital's Rule to solve similar limits?
4. What other factoring techniques can be used to simplify rational expressions?
5. What is the graphical interpretation of this type of limit?

**Tip**: For limits involving a zero denominator, carefully analyze if the limit tends to \( \infty \), \( -\infty \), or if it does not exist at all (DNE).

Find the one-sided limit \( L \) of \( \lim_{{x \to 2^-}} \frac{x^2 + 4x + 4}{x - 2} \) (if it exists). If an answer does not exist, enter DNE.

This detailed solution explains how to calculate the one-sided limit \( \lim_{{x \to 2^-}} \frac{x^2 + 4x + 4}{x - 2} \). Learn how to factor the quadratic expression and determine the limit as x approaches 2 from the left, leading to a result of negative infinity.

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