To evaluate the limit

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

let's proceed step by step.

Step 1: Factorize the numerator $x^4 - 9x^2 + 4x - 12$

The numerator is a polynomial of degree 4. We use grouping and substitution to simplify. Rewrite $x^4 - 9x^2 + 4x - 12$ as:

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

Notice that $x^4 - 9x^2$ is a difference of squares:

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

Thus, the numerator becomes:

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

Now, group terms involving $x - 3$:

$(x^2 - 3x)(x^2 + 3x) + 4(x - 3) = (x - 3)[(x^2 + 3x) + 4].$

Simplify inside the bracket:

$x^2 + 3x + 4.$

So, the numerator factors as:

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

Step 2: Simplify the expression

The denominator is $(x-3)(4-x)$. Therefore, the fraction becomes:

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

Cancel out the common factor $x - 3$ (valid as $x \to 3$):

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

Step 3: Evaluate the limit as $x \to 3$

Substitute $x = 3$ into the simplified expression:

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

Simplify the numerator and denominator:

$\text{Numerator: } 3^2 + 3(3) + 4 = 9 + 9 + 4 = 22,$ $\text{Denominator: } 4 - 3 = 1.$

Thus:

$\frac{x^2 + 3x + 4}{4 - x} = \frac{22}{1} = 22.$

Final Answer:

$\lim_{x \to 3} \frac{x^4 - 9x^2 + 4x - 12}{(x-3)(4-x)} = 22.$

Would you like further clarification or additional steps explained? Here are five related questions for deeper understanding:

1. How does the cancellation of terms work in limits, and why is it valid?
2. Can the numerator $x^4 - 9x^2 + 4x - 12$ be factored differently for other roots?
3. What is the role of the difference of squares in simplifying polynomial expressions?
4. Why is it important to verify if a term approaches 0 before canceling it in a fraction?
5. How would the limit change if the denominator were slightly altered?

Tip: Always check the domain of the function and ensure no division by zero before simplifying limits.