Math Problem Statement

What is the value of \(\lim_{x \to 3} \frac{x^4 - 9x^2 + 4x - 12}{(x - 3)(4 - x)}\)?

Solution

We are tasked to evaluate the limit:

limx3x49x2+4x12(x3)(4x).\lim_{x \to 3} \frac{x^4 - 9x^2 + 4x - 12}{(x - 3)(4 - x)}.

Step 1: Simplify the numerator

The numerator is x49x2+4x12x^4 - 9x^2 + 4x - 12. Let's try factoring or simplifying this polynomial.

Step 2: Direct substitution check

Substituting x=3x = 3 in the numerator and denominator results in 0/00/0, indicating an indeterminate form. We need to factorize and cancel common terms.

Step 3: Rewrite the denominator

The denominator is (x3)(4x)(x - 3)(4 - x). Notice that 4x=(x4)4 - x = -(x - 4), so the denominator becomes: (x3)(4x)=(x3)(x4).(x - 3)(4 - x) = -(x - 3)(x - 4).

Thus, the limit is now: limx3x49x2+4x12(x3)(x4).\lim_{x \to 3} \frac{x^4 - 9x^2 + 4x - 12}{-(x - 3)(x - 4)}.

Step 4: Factorize the numerator

We use synthetic or polynomial division to factor x3x - 3 out of the numerator x49x2+4x12x^4 - 9x^2 + 4x - 12.

After dividing, we find: x49x2+4x12=(x3)(x3+3x26x4).x^4 - 9x^2 + 4x - 12 = (x - 3)(x^3 + 3x^2 - 6x - 4).

Thus, the limit becomes: limx3(x3)(x3+3x26x4)(x3)(x4).\lim_{x \to 3} \frac{(x - 3)(x^3 + 3x^2 - 6x - 4)}{-(x - 3)(x - 4)}.

Step 5: Cancel x3x - 3 terms

Cancel the common factor x3x - 3: limx3x3+3x26x4(x4).\lim_{x \to 3} \frac{x^3 + 3x^2 - 6x - 4}{-(x - 4)}.

Step 6: Evaluate the simplified limit

Now substitute x=3x = 3 into the remaining expression: x3+3x26x4(x4)x=3=33+3(32)6(3)4(34).\frac{x^3 + 3x^2 - 6x - 4}{-(x - 4)} \Big|_{x = 3} = \frac{3^3 + 3(3^2) - 6(3) - 4}{-(3 - 4)}.

Calculate step-by-step:

  1. Numerator: 33+3(32)6(3)4=27+27184=32.3^3 + 3(3^2) - 6(3) - 4 = 27 + 27 - 18 - 4 = 32.
  2. Denominator: (34)=(1)=1.-(3 - 4) = -(-1) = 1.

So, the result is: 321=32.\frac{32}{1} = 32.

Final Answer:

32\boxed{32}

Would you like a detailed explanation of any step? Here are some related questions to explore:

  1. What is the general method for factoring polynomials with high degrees?
  2. How can we handle indeterminate forms in limits?
  3. What alternative methods could solve this limit (e.g., L'Hopital's Rule)?
  4. How does synthetic division work in factoring polynomials?
  5. How does changing the denominator sign affect the simplification process?

Tip: Always check for common factors in the numerator and denominator before applying substitution to simplify limits effectively.

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Math Problem Analysis

Mathematical Concepts

Limits
Polynomial Factorization
Indeterminate Forms
Simplification of Rational Functions

Formulas

Direct substitution in limits
Factorization of polynomials
Simplification of rational expressions

Theorems

Limit Theorem for Polynomial Functions
Cancellation Theorem

Suitable Grade Level

Grades 11-12