Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.
As x -> -2^+, f(x) -> __

Verified step by step guidance

1. Identify the vertical asymptote closest to x = -2, which is x = -16.
2. Since x = -2 is not a vertical asymptote, observe the behavior of the function as x approaches -2 from the right (x -> -2^+).
3. Notice that the graph does not have any discontinuities or asymptotes at x = -2, indicating that the function is continuous around this point.
4. Examine the graph to determine the y-value that f(x) approaches as x approaches -2 from the right.
5. Conclude that as x -> -2^+, f(x) approaches the y-value observed on the graph.

Key Concepts

Rational Functions

A rational function is a function represented by the ratio of two polynomials. It is typically expressed in the form f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the behavior of rational functions is crucial for analyzing their graphs, particularly in identifying asymptotic behavior and intercepts.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. There are two main types: vertical asymptotes, which occur where the function is undefined (typically where the denominator is zero), and horizontal asymptotes, which describe the behavior of the function as x approaches infinity or negative infinity. These concepts help in predicting the end behavior of rational functions.

Limits

Limits are fundamental in calculus and are used to describe the behavior of functions as they approach a certain point. In the context of rational functions, limits help determine the value that f(x) approaches as x approaches a specific value, such as -2 from the right (denoted as -2^+). This is essential for understanding the function's behavior near vertical asymptotes.