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

As $x\to \infty ,f\left(x\right)\to$_____

Verified step by step guidance

1. Identify the horizontal asymptote from the graph. The horizontal asymptote is the line that the function approaches as x approaches infinity or negative infinity.
2. From the graph, observe the horizontal dashed line labeled as the horizontal asymptote, which is at y = 17.
3. Recall that for rational functions, the horizontal asymptote represents the value that f(x) approaches as x approaches infinity (x → ∞) or negative infinity (x → -∞).
4. Therefore, as x → ∞, the function f(x) approaches the horizontal asymptote y = 17.
5. Write the conclusion: As x → ∞, f(x) → 17.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where a rational function's denominator is zero and the function approaches infinity or negative infinity. They represent values of x where the function is undefined and the graph shows a vertical line that the curve approaches but never crosses.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, the horizontal asymptote is a horizontal line y = c that the graph approaches, indicating the end behavior of the function.

End Behavior of Rational Functions

The end behavior of a rational function is determined by the degrees of the numerator and denominator polynomials. It shows how the function behaves as x approaches positive or negative infinity, often approaching a horizontal asymptote or increasing/decreasing without bound.