What is the horizontal asymptote of [tex]f(x)=\frac{-2 x}{x+1}[/tex]?
Answer (2)
Divide both the numerator and the denominator by x : x + 1 − 2 x = 1 + x 1 − 2 .
Evaluate the limit as x approaches infinity: lim x → ∞ 1 + x 1 − 2 = − 2 .
Evaluate the limit as x approaches negative infinity: lim x → − ∞ 1 + x 1 − 2 = − 2 .
The horizontal asymptote is y = − 2 , since the function approaches -2 as x goes to ± ∞ . y = − 2
Explanation
Understanding the Problem We are given the function f ( x ) = x + 1 − 2 x and asked to find its horizontal asymptote. The horizontal asymptote describes the behavior of the function as x approaches positive or negative infinity.
Finding the Limits To find the horizontal asymptote, we need to evaluate the limit of the function as x approaches infinity and negative infinity. That is, we need to find lim x → ∞ x + 1 − 2 x and lim x → − ∞ x + 1 − 2 x .
Dividing by x To evaluate these limits, we can divide both the numerator and the denominator by the highest power of x present, which in this case is x . This gives us: x → ∞ lim x + 1 − 2 x = x → ∞ lim 1 + x 1 − 2 x → − ∞ lim x + 1 − 2 x = x → − ∞ lim 1 + x 1 − 2
Evaluating the Limits As x approaches infinity or negative infinity, the term x 1 approaches 0. Therefore, we have: x → ∞ lim 1 + x 1 − 2 = 1 + 0 − 2 = − 2 x → − ∞ lim 1 + x 1 − 2 = 1 + 0 − 2 = − 2
Conclusion Since both limits are equal to -2, the horizontal asymptote of the function is y = − 2 .
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, representing the maximum safe concentration. Similarly, in economics, the growth of a company's revenue might approach a horizontal asymptote, indicating a saturation point beyond which further growth is limited. In environmental science, the spread of pollutants in a contained environment might also be modeled using functions with asymptotes, showing the maximum level of contamination.
The horizontal asymptote of the function f ( x ) = x + 1 − 2 x is y = − 2 . This is determined by evaluating the limits of the function as x approaches positive and negative infinity, both of which equal − 2 . Thus, the graph approaches this line as x goes to infinity or negative infinity.
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