Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 15
Capitolo 4, Problema 15

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

As x→1+, f(x)→x\(\to\)1^{+},\(\text{ }\)f(x)\(\to\) __

Guida verificata passo dopo passo
1
Identify the vertical asymptote near the value x = 1 on the graph. Notice that the graph has vertical asymptotes at x = -16 and x = -8, but none at x = 1, so the behavior near x = 1 is not influenced by a vertical asymptote.
Observe the behavior of the function f(x) as x approaches 1 from the right (x \(\to\) 1^+). Look at the graph closely near x = 1 and see whether the function values increase, decrease, or approach a specific number.
From the graph, as x approaches 1 from the right, the function values seem to approach a certain y-value. Determine if the function is going towards positive infinity, negative infinity, or a finite number.
Since the graph near x = 1 shows the function values approaching a finite number, identify that number by looking at the y-axis value the graph approaches.
Conclude that as x \(\to\) 1^+, f(x) approaches the finite value you identified from the graph.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
1m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity as x approaches a specific value. They represent values of x that make the denominator zero in a rational function, causing the function to be undefined. The graph shows the function's behavior near these lines, indicating limits from the left or right.
Video consigliato:
3:12
Determining Vertical Asymptotes

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. They indicate the value that the function approaches but does not necessarily reach. For rational functions, horizontal asymptotes are determined by comparing the degrees of the numerator and denominator polynomials.
Video consigliato:
4:48
Determining Horizontal Asymptotes

Limit Behavior Near Asymptotes

The limit of a function as x approaches a certain value from the right or left describes how the function behaves near that point. Near vertical asymptotes, the function may approach positive or negative infinity. Understanding this helps in completing statements about the function's behavior as x approaches specific values.
Video consigliato:
6:24
Introduction to Asymptotes
Pratica correlata
Domanda del libro di testo

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 3x2+16x<−53x^2+16x<−5

526
views
Domanda del libro di testo

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 6x2+x>1

485
views
Domanda del libro di testo

In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE>

f(x)=−x4+x2f(x)=−x^4+x^2

726
views
Domanda del libro di testo

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−x2−2x+8

1559
views
Domanda del libro di testo

Find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=−2(x−1)(x+2)2(x+5)3f(x)=-2(x-1)(x+2)^2(x+5)^3

826
views
Domanda del libro di testo

Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square root of w.

594
views