We are asked to evaluate the limit:

$\lim_{x \to 2} \frac{x^2 + 4x + 12}{x^2 + 9x + 14}.$

Let's proceed step by step.

Step 1: Plugging in the value of $x = 2$

First, we'll substitute $x = 2$ directly into the expression to check if this limit can be solved by simple substitution.

$\frac{(2)^2 + 4(2) + 12}{(2)^2 + 9(2) + 14} = \frac{4 + 8 + 12}{4 + 18 + 14} = \frac{24}{36} = \frac{2}{3}.$

Conclusion:

The limit exists and is equal to:

$\lim_{x \to 2} \frac{x^2 + 4x + 12}{x^2 + 9x + 14} = \frac{2}{3}.$

Let me know if you need more details on the steps!

Here are five related questions to explore:

1. What is the difference between a one-sided limit and a two-sided limit?
2. How do you evaluate limits involving infinity?
3. What are the key indeterminate forms that can arise when solving limits?
4. How can L'Hôpital's Rule help in evaluating limits?
5. What is the geometric interpretation of a limit?

Tip: Always check if a limit can be directly computed by substitution first, before applying more complex techniques!