What are the vertical and horizontal asymptotes of [tex]f(x)=\frac{2 x}{x-1}[/tex]?
A. horizontal asymptote at [tex]y=0[/tex], vertical asymptote at [tex]x=1[/tex]
B. horizontal asymptote at [tex]y=2[/tex], vertical asymptote at [tex]x=1[/tex]
C. horizontal asymptote at [tex]y=1[/tex], vertical asymptote at [tex]x=0[/tex]
D. horizontal asymptote at [tex]y=1[/tex], vertical asymptote at [tex]x=2[/tex]
Answer (2)
Find the vertical asymptote by setting the denominator equal to zero: x − 1 = 0 , so x = 1 .
Find the horizontal asymptote by evaluating the limit as x approaches infinity: lim x → ∞ x − 1 2 x = 2 .
Find the horizontal asymptote by evaluating the limit as x approaches negative infinity: lim x → − ∞ x − 1 2 x = 2 .
The vertical asymptote is x = 1 and the horizontal asymptote is y = 2 , so the answer is \boxed{horizontal asymptote at y=2 , v er t i c a l a sy m pt o t e a t x=1 } .
Explanation
Problem Analysis We are given the function f ( x ) = x − 1 2 x and asked to find its vertical and horizontal asymptotes.
Finding Vertical Asymptote To find the vertical asymptote, we look for values of x where the denominator is zero. So, we solve x − 1 = 0 , which gives x = 1 . The numerator at x = 1 is 2 ( 1 ) = 2 , which is non-zero. Thus, there is a vertical asymptote at x = 1 .
Finding Horizontal Asymptote To find the horizontal asymptote, we need to find the limit of the function as x approaches infinity and negative infinity.
Let's find lim x → ∞ x − 1 2 x . We can divide both the numerator and the denominator by x :
x → ∞ lim x − 1 2 x = x → ∞ lim 1 − x 1 2 As x approaches infinity, x 1 approaches 0. Therefore, x → ∞ lim 1 − x 1 2 = 1 − 0 2 = 2 Similarly, let's find lim x → − ∞ x − 1 2 x . Dividing both the numerator and the denominator by x :
x → − ∞ lim x − 1 2 x = x → − ∞ lim 1 − x 1 2 As x approaches negative infinity, x 1 approaches 0. Therefore, x → − ∞ lim 1 − x 1 2 = 1 − 0 2 = 2 Since both limits are equal to 2, there is a horizontal asymptote at y = 2 .
Conclusion Therefore, the vertical asymptote is at x = 1 and the horizontal asymptote is at y = 2 .
Examples
Understanding asymptotes is crucial in various real-world applications. For instance, in pharmacology, drug concentration in the bloodstream often follows a curve that approaches a horizontal asymptote, representing the maximum safe concentration. Similarly, in economics, cost functions may approach a horizontal asymptote, indicating the minimum cost achievable with increasing production. In physics, the velocity of an object under certain conditions might approach a horizontal asymptote, representing its terminal velocity. These examples highlight how asymptotes help model and predict limits in diverse fields.
The function f ( x ) = x − 1 2 x has a vertical asymptote at x = 1 and a horizontal asymptote at y = 2 . The correct answer is option B.
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