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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 19
Capitolo 4, Problema 19

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.
Graph of a rational function with vertical asymptotes at x = -2 and x = 1, and horizontal asymptote at y = 1.
As x→∞, f(x)→x\(\to\)\(\infty\),\(\text{ }\)f(x)\(\to\)_____

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

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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.
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Determining Vertical Asymptotes

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.
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Determining Horizontal Asymptotes

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.
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End Behavior of Polynomial Functions
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