Which statement describes the behavior of the function [tex]f(x)=\frac{2 x}{1-x^2}[/tex]?
A. The graph approaches -2 as [tex]x[/tex] approaches infinity.
B. The graph approaches 0 as [tex]x[/tex] approaches infinity.
C. The graph approaches 1 as [tex]x[/tex] approaches infinity.
D. The graph approaches 2 as [tex]x[/tex] approaches infinity.
Answer (2)
Analyze the function f ( x ) = 1 − x 2 2 x as x approaches infinity.
Simplify the function to − x 2 for large x .
Evaluate the limit: lim x → ∞ − x 2 = 0 .
Conclude that the graph approaches 0 as x approaches infinity. 0
Explanation
Problem Analysis We are asked to determine the behavior of the function f ( x ) = 1 − x 2 2 x as x approaches infinity. In other words, we need to find the limit of the function as x goes to infinity.
Simplifying the Function To find the limit of the function f ( x ) as x approaches infinity, we can analyze the expression 1 − x 2 2 x . As x becomes very large, the term x 2 in the denominator will dominate the constant term 1. Thus, the function behaves like − x 2 2 x = − x 2 .
Evaluating the Limit Now, we can evaluate the limit: x → ∞ lim 1 − x 2 2 x = x → ∞ lim − x 2 As x approaches infinity, the fraction x 2 approaches 0. Therefore, the limit is 0.
Conclusion The graph of the function f ( x ) = 1 − x 2 2 x approaches 0 as x approaches infinity.
Examples
Understanding the behavior of functions as their input approaches infinity is crucial in many fields. For example, in physics, it can help model the decay of radioactive substances over long periods. In economics, it can predict the long-term trends of market prices. In computer science, it can analyze the efficiency of algorithms as the input size grows infinitely large. This concept is also used in engineering to design stable systems that can handle extreme conditions.
The behavior of the function f ( x ) = 1 − x 2 2 x as x approaches infinity can be analyzed by simplifying the function and finding the limit, which equals 0. Thus, the graph approaches 0 as x approaches infinity. The correct answer is option B.
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