Describe the end behavior of the following polynomial function using limit notation.
The Correct Answer and Explanation is :
To analyze the end behavior of a polynomial function using limit notation, we focus on the function’s leading term—the term with the highest degree—since it predominantly influences the function’s behavior as ( x ) approaches positive or negative infinity.
General Approach:
- Identify the Leading Term: Examine the polynomial and determine the term with the highest exponent.
- Determine the Degree and Leading Coefficient: The degree is the exponent of the leading term, and the leading coefficient is the constant multiplying this term.
- Analyze the End Behavior:
- If the degree is even and the leading coefficient is positive: As ( x \to \infty ), ( f(x) \to \infty ); as ( x \to -\infty ), ( f(x) \to \infty ).
- If the degree is even and the leading coefficient is negative: As ( x \to \infty ), ( f(x) \to -\infty ); as ( x \to -\infty ), ( f(x) \to -\infty ).
- If the degree is odd and the leading coefficient is positive: As ( x \to \infty ), ( f(x) \to \infty ); as ( x \to -\infty ), ( f(x) \to -\infty ).
- If the degree is odd and the leading coefficient is negative: As ( x \to \infty ), ( f(x) \to -\infty ); as ( x \to -\infty ), ( f(x) \to \infty ).
Example:
Consider the polynomial function:
[ f(x) = 2x^4 – 3x^3 + x – 5 ]
- Leading Term: The term with the highest degree is ( 2x^4 ).
- Degree: The degree is 4 (even).
- Leading Coefficient: The leading coefficient is 2 (positive).
End Behavior Analysis:
- As ( x \to \infty ): The leading term ( 2x^4 ) dominates, and since it is positive, ( f(x) \to \infty ).
- As ( x \to -\infty ): The leading term ( 2x^4 ) remains positive (since raising a negative number to an even power results in a positive number), so ( f(x) \to \infty ).
Limit Notation:
- As ( x \to \infty ), ( \lim_{x \to \infty} f(x) = \infty ).
- As ( x \to -\infty ), ( \lim_{x \to -\infty} f(x) = \infty ).
This analysis shows that the function increases without bound as ( x ) moves away from zero in both the positive and negative directions.
For a more detailed explanation and additional examples, you might find the following video helpful: