Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
As x -> -∞, f(x) -> __

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1. Identify the horizontal asymptote of the function, which is y = 0.
2. As x approaches negative infinity, observe the behavior of the function relative to the horizontal asymptote.
3. Since the horizontal asymptote is y = 0, the function f(x) will approach 0 as x approaches negative infinity.
4. The graph shows that as x moves towards negative infinity, the function values get closer to the horizontal asymptote y = 0.
5. Conclude that as x -> -∞, f(x) -> 0.

Key Concepts

Rational Functions

A rational function is a function represented by the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding rational functions is crucial for analyzing their behavior, including identifying asymptotes and intercepts, which are key to interpreting their graphs.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. Vertical asymptotes occur where the function is undefined, typically where the denominator equals zero, while horizontal asymptotes indicate the behavior of the function as x approaches infinity or negative infinity. In this case, the horizontal asymptote at y=0 suggests that as x approaches negative infinity, f(x) approaches 0.

End Behavior of Functions

The end behavior of a function describes how the function behaves as the input values approach positive or negative infinity. For rational functions, this behavior is often determined by the degrees of the numerator and denominator polynomials. In this case, as x approaches negative infinity, the function approaches the horizontal asymptote, indicating that f(x) approaches 0.