The problem presented is the following limit:

$\lim_{{x \to 2}} \frac{x^2 + 4x + 12}{x^2 + 9x + 14} = 0$

To solve this, let's first evaluate the limit by direct substitution of $x = 2$.

Step 1: Substitute $x = 2$ into the numerator and denominator.

Numerator:

$x^2 + 4x + 12 = 2^2 + 4(2) + 12 = 4 + 8 + 12 = 24$

Denominator:

$x^2 + 9x + 14 = 2^2 + 9(2) + 14 = 4 + 18 + 14 = 36$

Step 2: Evaluate the limit.

Now substitute these values back into the fraction:

$\frac{24}{36} = \frac{2}{3}$

This is not equal to 0. Hence, the equation as written is incorrect if we expect the result to be 0.

Conclusion:

The limit, as evaluated, equals $\frac{2}{3}$, not 0. It seems there might be a mistake in the setup of the original equation.

Would you like more details or clarifications?

Here are 5 related questions:

1. What happens if the denominator becomes 0 at $x = 2$?
2. How would factoring help solve this limit if it were indeterminate?
3. What are the conditions for applying L'Hopital's Rule?
4. Can limits be solved graphically for better understanding?
5. How do you handle limits approaching infinity?

Tip: Always try direct substitution in limit problems first before applying other methods like factoring or L'Hopital's Rule.