We are tasked with evaluating the limit:

$\lim_{x \to 3} \frac{x^4 - 9x^2 + 4x - 12}{(x-3)(4-x)}.$

Step 1: Simplify the denominator

The denominator is the product $(x - 3)(4 - x)$. Notice that we can rewrite $4 - x$ as $-(x - 4)$, so:

$(x - 3)(4 - x) = (x - 3)(-(x - 4)) = -(x - 3)(x - 4).$

Thus, the original limit becomes:

$\lim_{x \to 3} \frac{x^4 - 9x^2 + 4x - 12}{-(x - 3)(x - 4)}.$

Step 2: Check for direct substitution

Now, let's try substituting $x = 3$ directly into the expression to see if we can determine the limit or if the expression becomes indeterminate:

1. In the numerator: $x^4 - 9x^2 + 4x - 12 \text{ at } x = 3 \Rightarrow 3^4 - 9(3^2) + 4(3) - 12 = 81 - 81 + 12 - 12 = 0.$

2. In the denominator: $-(x - 3)(x - 4) \text{ at } x = 3 \Rightarrow -(3 - 3)(3 - 4) = -(0)(-1) = 0.$

Thus, direct substitution leads to the indeterminate form $\frac{0}{0}$, so we need to simplify further.

Step 3: Factor the numerator

We now try factoring the numerator $x^4 - 9x^2 + 4x - 12$. First, let's rewrite it in terms of lower-degree terms. Notice that this is a quartic polynomial, and we look for a factorization of the form:

$x^4 - 9x^2 + 4x - 12 = (x^2 + ax + b)(x^2 + cx + d).$

However, factoring this directly might be complicated. Let's try to use polynomial division instead.

Step 4: Polynomial division

We will divide the numerator $x^4 - 9x^2 + 4x - 12$ by $(x - 3)$. After performing synthetic or long division, we obtain:

$x^4 - 9x^2 + 4x - 12 = (x - 3)(x^3 + 3x^2 - 12x + 4).$

Step 5: Simplify the limit expression

Now we can simplify the original limit expression by canceling out the factor $(x - 3)$ from the numerator and the denominator:

$\lim_{x \to 3} \frac{(x - 3)(x^3 + 3x^2 - 12x + 4)}{-(x - 3)(x - 4)} = \lim_{x \to 3} \frac{x^3 + 3x^2 - 12x + 4}{-(x - 4)}.$

Since we canceled the $(x - 3)$ term, the limit is no longer indeterminate. Now we can substitute $x = 3$ directly into the simplified expression:

1. In the numerator: $x^3 + 3x^2 - 12x + 4 \text{ at } x = 3 \Rightarrow 3^3 + 3(3^2) - 12(3) + 4 = 27 + 27 - 36 + 4 = 22.$

2. In the denominator: $-(x - 4) \text{ at } x = 3 \Rightarrow -(3 - 4) = 1.$

Thus, the limit is:

$\frac{22}{1} = 22.$

Final Answer:

The value of the limit is $\boxed{22}$.

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Would you like further details on any step, or do you have any questions about this process? Here are a few related questions to explore:

1. How can you apply polynomial division to simplify limits involving higher degree polynomials?
2. What techniques are used to handle indeterminate forms like $\frac{0}{0}$?
3. How do you factor quartic polynomials efficiently?
4. Why is it important to identify and cancel common factors in limits?
5. How can you use synthetic division to divide polynomials?

Tip: When encountering an indeterminate form, such as $\frac{0}{0}$, try factoring the numerator and denominator or using polynomial division to simplify the expression.