Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

5. Rational Functions

Asymptotes

College Algebra

5. Rational Functions

Asymptotes

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1:11 minutes

Problem 15

Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.
As x -> 1^+, f(x) -> __

Verified step by step guidance

Identify the behavior of the function as x approaches 1 from the right (x -> 1^+).

Observe the graph near x = 1 to determine the direction in which the function is heading.

Note that the graph does not show any vertical asymptote at x = 1, so the function is continuous around this point.

Check if the function is increasing or decreasing as it approaches x = 1 from the right.

Conclude the behavior of f(x) as x approaches 1 from the right based on the observed trend in the graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 discontinuities, which are key to interpreting their graphs.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur where the function is undefined, typically where the denominator equals zero. Horizontal asymptotes indicate the behavior of the function as x approaches infinity or negative infinity, providing insight into the function's end behavior.

Limits

Limits describe the behavior of a function as the input approaches a certain value. In this context, evaluating the limit as x approaches a specific value (like 1 from the right) helps determine the corresponding output of the function. This concept is essential for understanding continuity and the behavior of rational functions near their asymptotes.