To evaluate the given limit

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

let's proceed step by step:

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Step 1: Simplify the denominator

The denominator is $(x-3)(4-x)$. Notice that:

$4 - x = -(x - 4),$

so the denominator becomes:

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

Thus, the expression becomes:

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

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Step 2: Analyze the numerator

The numerator is $x^4 - 9x^2 + 4x - 12$. Let's check for factors involving $(x-3)$ and $(x-4)$ by substituting $x = 3$ and $x = 4$:

• At $x = 3$:

$3^4 - 9(3^2) + 4(3) - 12 = 81 - 81 + 12 - 12 = 0.$

Thus, $x-3$ is a factor.

• At $x = 4$:

$4^4 - 9(4^2) + 4(4) - 12 = 256 - 144 + 16 - 12 = 0.$

Thus, $x-4$ is also a factor.

This means the numerator has $(x-3)(x-4)$ as factors. To factorize the numerator completely, we perform synthetic division.

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Step 3: Factorize the numerator

We divide $x^4 - 9x^2 + 4x - 12$ by $(x-3)(x-4)$. Start by dividing by $(x-3)$, followed by $(x-4)$.

Synthetic division by $x-3$:

Dividing $x^4 - 9x^2 + 4x - 12$ by $x-3$ gives:

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

Synthetic division by $x-4$:

Dividing $x^3 + 3x^2 - 3x + 4$ by $x-4$ gives:

$x^3 + 3x^2 - 3x + 4 = (x-4)(x^2 + 7x + 1).$

Thus, the full factorization of the numerator is:

$x^4 - 9x^2 + 4x - 12 = (x-3)(x-4)(x^2 + 7x + 1).$

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Step 4: Simplify the limit expression

Substitute the factorized numerator into the expression:

$\frac{x^4 - 9x^2 + 4x - 12}{-(x-3)(x-4)} = \frac{(x-3)(x-4)(x^2 + 7x + 1)}{-(x-3)(x-4)}.$

Cancel the common factors $(x-3)$ and $(x-4)$, leaving:

$\frac{x^2 + 7x + 1}{-1}.$

Simplify further:

$-(x^2 + 7x + 1) = -x^2 - 7x - 1.$

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Step 5: Evaluate the limit

Since there are no more problematic terms, directly substitute $x = 3$ into $-x^2 - 7x - 1$:

$- (3^2) - 7(3) - 1 = -9 - 21 - 1 = -31.$

Thus, the value of the limit is:

$\boxed{-31}.$

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Do you want further clarification or a detailed explanation of any step? Here are some related questions to consider:

1. How is synthetic division performed for polynomial factorization?
2. Why is it important to analyze $x = 3$ and $x = 4$ for the numerator?
3. How does canceling common factors simplify a limit problem?
4. What are common techniques for solving limits involving rational functions?
5. Can this problem be solved using L’Hôpital’s Rule?

Tip: Always check for common factors in the numerator and denominator before directly substituting to avoid indeterminate forms like