The given mathematical expression is:

$\lim_{{x \to -2}} \frac{x^2 - 4x + 12}{x + 2}$

Let's solve this step-by-step:

Step 1: Direct Substitution

First, we attempt to substitute $x = -2$ into the expression.

$f(x) = \frac{x^2 - 4x + 12}{x + 2}$

Substitute $x = -2$:

$\frac{(-2)^2 - 4(-2) + 12}{-2 + 2} = \frac{4 + 8 + 12}{0} = \frac{24}{0}$

Since the denominator becomes 0, this suggests a vertical asymptote or an undefined limit at $x = -2$. However, we can analyze the function more carefully to confirm the behavior of the limit.

Step 2: Behavior Around $x = -2$

We see that direct substitution results in division by zero, meaning the function has a discontinuity at $x = -2$. There is no factorization that simplifies the expression, so the limit does not exist in the standard sense because the result tends to infinity.

Conclusion:

The limit is undefined because the denominator tends to zero while the numerator remains finite, indicating a vertical asymptote at $x = -2$.

Would you like further details on this or have any questions?

Here are five related questions to explore:

1. How do you determine if a function has a vertical asymptote?
2. What does it mean when a limit tends to infinity?
3. How can L'Hopital's Rule be applied to limits involving indeterminate forms?
4. What are the common techniques for solving limits involving rational functions?
5. How can we graphically interpret the behavior of a function near a point of discontinuity?

Tip: Always check for possible factorizations or other algebraic simplifications before concluding that a limit does not exist.