Which of the following describes the end behavior of [tex]f(x)=\frac{2 x}{3 x^2-3}[/tex]?
The graph approaches 0 as [tex]x[/tex] approaches infinity.
The graph approaches 0 as [tex]x[/tex] approaches negative infinity.
The graph approaches [tex]2 / 3[/tex] as [tex]x[/tex] approaches infinity.
The graph approaches -1 as [tex]x[/tex] approaches negative infinity.

Find the limit of the function f ( x ) = 3 x 2 − 3 2 x ​ as x approaches infinity.
Divide the numerator and denominator by x 2 and simplify the expression.
Evaluate the limit as x approaches infinity, which results in 0.
Similarly, find the limit as x approaches negative infinity, which also results in 0. Therefore, the graph approaches 0 as x approaches both infinity and negative infinity. 0 ​

Explanation

Problem Analysis We are asked to determine the end behavior of the function f ( x ) = 3 x 2 − 3 2 x ​ as x approaches infinity and negative infinity. This means we need to find the limits of the function as x goes to ∞ and − ∞ .

Limit as x approaches infinity To find the limit as x approaches infinity, we can analyze the function's behavior when x becomes very large. We have: x → ∞ lim ​ 3 x 2 − 3 2 x ​ We can divide both the numerator and the denominator by the highest power of x in the denominator, which is x 2 :
x → ∞ lim ​ x 2 3 x 2 ​ − x 2 3 ​ x 2 2 x ​ ​ = x → ∞ lim ​ 3 − x 2 3 ​ x 2 ​ ​ As x approaches infinity, x 2 ​ approaches 0 and x 2 3 ​ approaches 0. Therefore, the limit is: 3 − 0 0 ​ = 0

Limit as x approaches negative infinity Similarly, to find the limit as x approaches negative infinity, we have: x → − ∞ lim ​ 3 x 2 − 3 2 x ​ We divide both the numerator and the denominator by x 2 :
x → − ∞ lim ​ x 2 3 x 2 ​ − x 2 3 ​ x 2 2 x ​ ​ = x → − ∞ lim ​ 3 − x 2 3 ​ x 2 ​ ​ As x approaches negative infinity, x 2 ​ approaches 0 and x 2 3 ​ approaches 0. Therefore, the limit is: 3 − 0 0 ​ = 0

Conclusion Thus, the graph approaches 0 as x approaches infinity and as x approaches negative infinity.


Examples
Understanding the end behavior of functions is crucial in many real-world applications. For example, in physics, it can help predict the long-term behavior of a system, such as the decay of a radioactive substance or the motion of a damped oscillator. In economics, it can be used to model the long-term growth of a company or the stability of a market. By analyzing the limits of functions as the input approaches infinity, we can gain valuable insights into the behavior of these systems over extended periods.

Answered by GinnyAnswer | 2025-07-05