Math Problem Statement

If f(x) = { x³ + x² - 16x + 20 / (x-2)², x ≠ 2 k, x = 2 } is continuous at x = 2, find k.

Solution

To determine the value of kk such that the function f(x)f(x) is continuous at x=2x = 2, we must ensure that the limit of f(x)f(x) as x2x \to 2 equals f(2)=kf(2) = k.

Step 1: Write the continuity condition

For f(x)f(x) to be continuous at x=2x = 2,

limx2f(x)=f(2)=k\lim_{x \to 2} f(x) = f(2) = k

Here, the function f(x)f(x) for x2x \neq 2 is given as:

f(x)=x3+x216x+20(x2)2f(x) = \frac{x^3 + x^2 - 16x + 20}{(x-2)^2}

We need to compute limx2f(x)\lim_{x \to 2} f(x).

Step 2: Simplify the numerator

The numerator of f(x)f(x) is x3+x216x+20x^3 + x^2 - 16x + 20. To evaluate the limit, note that the denominator (x2)2(x-2)^2 becomes 0 at x=2x = 2, so we factorize the numerator.

Factorization:

x3+x216x+20=(x2)(x2+3x10)x^3 + x^2 - 16x + 20 = (x-2)(x^2 + 3x - 10)

The quadratic x2+3x10x^2 + 3x - 10 can further be factored as:

x2+3x10=(x2)(x+5)x^2 + 3x - 10 = (x-2)(x+5)

Thus, the full factorization is:

x3+x216x+20=(x2)2(x+5)x^3 + x^2 - 16x + 20 = (x-2)^2(x+5)

Step 3: Rewrite f(x)f(x)

Substituting the factorization into the expression for f(x)f(x), we get:

f(x)=(x2)2(x+5)(x2)2f(x) = \frac{(x-2)^2(x+5)}{(x-2)^2}

For x2x \neq 2, cancel the (x2)2(x-2)^2 terms:

f(x)=x+5f(x) = x + 5

Step 4: Evaluate the limit as x2x \to 2

As x2x \to 2, f(x)2+5=7f(x) \to 2 + 5 = 7. Therefore:

limx2f(x)=7\lim_{x \to 2} f(x) = 7

Step 5: Determine kk

For f(x)f(x) to be continuous at x=2x = 2, we set:

k=limx2f(x)=7k = \lim_{x \to 2} f(x) = 7

Final Answer:

k=7\boxed{k = 7}

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Related Questions:

  1. What are the steps for checking continuity at a given point?
  2. How do you factorize cubic polynomials efficiently?
  3. What is the significance of removing removable discontinuities in functions?
  4. How do you compute limits involving indeterminate forms like 0/00/0?
  5. Can you explain how to verify continuity for piecewise functions?

Tip:

Always factorize the numerator when dealing with rational functions where the denominator becomes zero at a point of interest—it often simplifies the computation of limits.

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Math Problem Analysis

Mathematical Concepts

Continuity
Limits
Polynomial Factorization
Piecewise Functions

Formulas

Limit formula: lim(x → c) f(x) = f(c)
Factorization of cubic polynomials
Limit simplification for rational functions

Theorems

Definition of Continuity
Factorization Theorem

Suitable Grade Level

Grades 11-12