Find the horizontal asymptote of the function $\frac{(x-1)(x-3)^2(x+1)^2}{(x-2)(x+2)(x-1)(x+3)}$.
A. $y=1$
B. none
C. $x=0$
D. $y=0$
Answer (2)
Simplify the rational function by canceling common factors.
Expand the numerator and denominator to find their degrees.
Compare the degrees of the numerator and denominator.
Since the degree of the numerator (4) is greater than the degree of the denominator (3), there is no horizontal asymptote.
The final answer is none .
Explanation
Problem Analysis We are given the function f ( x ) = ( x − 2 ) ( x + 2 ) ( x − 1 ) ( x + 3 ) ( x − 1 ) ( x − 3 ) 2 ( x + 1 ) 2 and asked to find its horizontal asymptote.
Simplifying the Function First, we simplify the function by canceling the common factor ( x − 1 ) from the numerator and the denominator: f ( x ) = ( x − 2 ) ( x + 2 ) ( x + 3 ) ( x − 3 ) 2 ( x + 1 ) 2 .
Expanding Numerator and Denominator Next, we expand the numerator and the denominator to determine the degrees of the polynomials: ( x − 3 ) 2 ( x + 1 ) 2 = ( x 2 − 6 x + 9 ) ( x 2 + 2 x + 1 ) = x 4 + 2 x 3 + x 2 − 6 x 3 − 12 x 2 − 6 x + 9 x 2 + 18 x + 9 = x 4 − 4 x 3 − 2 x 2 + 12 x + 9 ( x − 2 ) ( x + 2 ) ( x + 3 ) = ( x 2 − 4 ) ( x + 3 ) = x 3 + 3 x 2 − 4 x − 12 So, we have f ( x ) = x 3 + 3 x 2 − 4 x − 12 x 4 − 4 x 3 − 2 x 2 + 12 x + 9 .
Determining the Asymptote The degree of the numerator is 4, and the degree of the denominator is 3. Since the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote. Instead, it has an oblique asymptote.
Final Answer Therefore, the horizontal asymptote of the given function is none.
Examples
Understanding asymptotes is crucial in fields like physics and engineering, where functions model real-world phenomena. For instance, in circuit analysis, the impedance of a circuit can be represented as a function of frequency. Identifying asymptotes helps engineers predict the circuit's behavior at extreme frequencies, ensuring stable and efficient operation. This knowledge is also vital in designing control systems, where asymptotes indicate system stability and response characteristics.
The horizontal asymptote of the function f ( x ) = ( x − 2 ) ( x + 2 ) ( x − 1 ) ( x + 3 ) ( x − 1 ) ( x − 3 ) 2 ( x + 1 ) 2 is none, as the degree of the numerator is greater than that of the denominator. This results in the function not having a horizontal asymptote. The correct answer is option B: "none".
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