What are the vertical and horizontal asymptotes for the function [tex]f(x)=\frac{3 x^2}{x^2-4}[/tex]?
A. horizontal asymptote: [tex]y=-2, y=2[/tex]
vertical asymptote: [tex]x=3[/tex]
B. horizontal asymptote: [tex]y=-4, y=1[/tex]
vertical asymptote: [tex]x=3[/tex]
C. horizontal asymptote: [tex]y=3[/tex]
vertical asymptote: [tex]x=-4, x=1[/tex]
D. horizontal asymptote: [tex]y=3[/tex]
vertical asymptote: [tex]x=-2, x=2[/tex]
Answer (2)
Find the vertical asymptotes by setting the denominator equal to zero: x 2 − 4 = 0 , which gives x = − 2 and x = 2 .
Determine the horizontal asymptote by evaluating the limit as x approaches infinity. Since the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients: y = 1 3 = 3 .
The vertical asymptotes are x = − 2 and x = 2 .
The horizontal asymptote is y = 3 .
Explanation
Problem Analysis We are given the function f ( x ) = x 2 − 4 3 x 2 and we need to find its vertical and horizontal asymptotes.
Finding Vertical Asymptotes To find the vertical asymptotes, we need to determine where the denominator of the rational function is equal to zero. So, we solve the equation x 2 − 4 = 0 .
Solving for Vertical Asymptotes x 2 − 4 = 0 can be factored as ( x − 2 ) ( x + 2 ) = 0 . Thus, the solutions are x = 2 and x = − 2 . These are the vertical asymptotes.
Finding Horizontal Asymptotes To find the horizontal asymptote, we examine the limit of the function as x approaches infinity. Since the degree of the numerator and the denominator are the same (both are 2), the horizontal asymptote is the ratio of the leading coefficients.
Calculating the Limit The limit as x approaches infinity is: x → ∞ lim x 2 − 4 3 x 2 = x → ∞ lim 1 − x 2 4 3 = 1 − 0 3 = 3 Thus, the horizontal asymptote is y = 3 .
Final Answer Therefore, the vertical asymptotes are x = − 2 and x = 2 , and the horizontal asymptote is y = 3 .
Examples
Understanding asymptotes is crucial in various fields. For instance, in epidemiology, when modeling the spread of a disease, the horizontal asymptote can represent the maximum number of people who will be infected. In economics, it can represent the saturation point of a market. In physics, asymptotes appear when describing the behavior of fields near point charges or masses. Knowing how to find asymptotes helps in predicting the long-term behavior of these models.
The vertical asymptotes of the function f ( x ) = x 2 − 4 3 x 2 are at x = − 2 and x = 2 , while the horizontal asymptote is at y = 3 . Therefore, the answer is option D: horizontal asymptote: y = 3 and vertical asymptote: x = − 2 , x = 2 .
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