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 such that the function is continuous at , we must ensure that the limit of as equals .
Step 1: Write the continuity condition
For to be continuous at ,
Here, the function for is given as:
We need to compute .
Step 2: Simplify the numerator
The numerator of is . To evaluate the limit, note that the denominator becomes 0 at , so we factorize the numerator.
Factorization:
The quadratic can further be factored as:
Thus, the full factorization is:
Step 3: Rewrite
Substituting the factorization into the expression for , we get:
For , cancel the terms:
Step 4: Evaluate the limit as
As , . Therefore:
Step 5: Determine
For to be continuous at , we set:
Final Answer:
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Related Questions:
- What are the steps for checking continuity at a given point?
- How do you factorize cubic polynomials efficiently?
- What is the significance of removing removable discontinuities in functions?
- How do you compute limits involving indeterminate forms like ?
- 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
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