What is the horizontal asymptote of the function [tex]$f(x)=\frac{(x-2)}{(x-3)^2}$[/tex]?
A. [tex]$y=0$[/tex]
B. [tex]$y=1$[/tex]
C. [tex]$y=2$[/tex]
D. [tex]$y=3$[/tex]
Answer (2)
To find the horizontal asymptote of f ( x ) = ( x − 3 ) 2 ( x − 2 ) , we examine the limit as x approaches ± ∞ .
Expand the denominator: f ( x ) = x 2 − 6 x + 9 x − 2 .
Divide the numerator and denominator by x 2 : f ( x ) = 1 − x 6 + x 2 9 x 1 − x 2 2 .
Evaluate the limit as x approaches ± ∞ , which gives y = 0 , so the horizontal asymptote is y = 0 .
Explanation
Understanding Horizontal Asymptotes We are asked to find the horizontal asymptote of the function f ( x ) = ( x − 3 ) 2 ( x − 2 ) . The horizontal asymptote is the value that the function approaches as x goes to positive or negative infinity.
Finding the Limits To find the horizontal asymptote, we need to evaluate the limits of the function as x approaches infinity and negative infinity. That is, we need to find lim x → ∞ ( x − 3 ) 2 ( x − 2 ) and lim x → − ∞ ( x − 3 ) 2 ( x − 2 ) .
Expanding the Denominator Let's first expand the denominator: ( x − 3 ) 2 = x 2 − 6 x + 9 . So, the function becomes f ( x ) = x 2 − 6 x + 9 x − 2 .
Dividing by the Highest Power of x To evaluate the limit as x approaches infinity, we divide both the numerator and the denominator by the highest power of x in the denominator, which is x 2 . This gives us:
x 2 − 6 x + 9 x − 2 = x 2 x 2 − x 2 6 x + x 2 9 x 2 x − x 2 2 = 1 − x 6 + x 2 9 x 1 − x 2 2
Evaluating the Limit Now, as x approaches infinity, the terms x 1 , x 2 2 , x 6 , and x 2 9 all approach 0. Therefore, the limit as x approaches infinity is:
x → ∞ lim 1 − x 6 + x 2 9 x 1 − x 2 2 = 1 − 0 + 0 0 − 0 = 1 0 = 0
Similarly, as x approaches negative infinity, the same terms approach 0, so the limit as x approaches negative infinity is also 0.
Conclusion Since lim x → ∞ f ( x ) = 0 and lim x → − ∞ f ( x ) = 0 , the horizontal asymptote is y = 0 .
Examples
Understanding horizontal asymptotes is crucial in various real-world applications. For instance, in pharmacology, the concentration of a drug in the bloodstream over time can be modeled by a function with a horizontal asymptote. The asymptote represents the long-term concentration of the drug, indicating the level at which the body eliminates the drug as quickly as it's administered, achieving a steady state. This helps doctors determine appropriate dosages to maintain therapeutic levels without causing toxicity.
The horizontal asymptote of the function f ( x ) = ( x − 3 ) 2 ( x − 2 ) is found by evaluating the limits as x approaches infinity and negative infinity, both yielding a value of 0. Therefore, the horizontal asymptote is y = 0 and the correct answer is option A.
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