Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As x -> ∞, f(x) -> __

Verified step by step guidance

1. Identify the horizontal asymptote from the graph, which is the line that the function approaches as x approaches infinity.
2. From the graph, observe that the horizontal asymptote is given by the line y = 17.
3. Recall that for rational functions, the horizontal asymptote represents the value that f(x) approaches as x tends to positive or negative infinity.
4. Therefore, as x → ∞, f(x) approaches the horizontal asymptote y = 17.
5. Conclude that the limit of f(x) as x approaches infinity is 17.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a specific value. They indicate values of x where the function is undefined, often due to division by zero in rational functions. In the graph, vertical asymptotes are shown at x = -16 and x = -8.

Horizontal Asymptotes

A horizontal asymptote represents the value that a function approaches as x approaches infinity or negative infinity. It shows the end behavior of the function. In this graph, the horizontal asymptote is y = 17, meaning as x → ∞, f(x) approaches 17.

End Behavior of Rational Functions

The end behavior describes how a function behaves as x becomes very large or very small. For rational functions, this is often determined by comparing the degrees of the numerator and denominator polynomials, which helps identify horizontal asymptotes and limits at infinity.