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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.5.55b
Chapter 2, Problem 2.5.55b

Complete the following steps for the given functions. 


b. Find the vertical asymptotes of ff (if any).


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

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1
Identify the function given: \( f(x) = \frac{4x^3 + 4x^2 + 7x + 4}{x^2 + 1} \). This is a rational function, which is a ratio of two polynomials.
To find vertical asymptotes, we need to determine where the denominator is equal to zero, as these are the points where the function is undefined.
Set the denominator equal to zero: \( x^2 + 1 = 0 \).
Solve the equation \( x^2 + 1 = 0 \) for \( x \). Notice that \( x^2 = -1 \), which has no real solutions since the square of a real number cannot be negative.
Conclude that there are no vertical asymptotes for the function \( f(x) = \frac{4x^3 + 4x^2 + 7x + 4}{x^2 + 1} \) because the denominator does not have any real roots.

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Key Concepts

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

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator approaches zero while the numerator does not. These points indicate where the function's value tends to infinity or negative infinity, leading to a discontinuity. To find vertical asymptotes, set the denominator equal to zero and solve for the variable, ensuring that the numerator does not also equal zero at those points.
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Asymptotes of Hyperbolas

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 the behavior of rational functions is crucial for analyzing their asymptotic behavior, including identifying vertical and horizontal asymptotes, as well as points of discontinuity.
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Intro to Rational Functions

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. They play a significant role in calculus, particularly in determining the behavior of rational functions. The degree of the polynomial affects the function's end behavior and can influence the presence of asymptotes.
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Related Practice
Textbook Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time. 


c. s(t)=40 sin 2t at t=0

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Textbook Question

Complete the following steps for the given functions. 


b. Find the vertical asymptotes of ff (if any).


f(x)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

363
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Textbook Question

Complete the following steps for the given functions. 


c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

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Textbook Question

Determine the following limits.


b. limx→2x−2x5−4x3{\(\displaystyle\]\lim\)_{x\(\to\)2}}\(\frac{x-2}{x^5-4x^3}\)

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Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

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Textbook Question

Use the graph of gg in the figure to find the following values or state that they do not exist. <IMAGE>

limx→0g(x){\(\displaystyle\]\lim\)_{x\(\to\)0}g\(\left\)(x\(\right\))}

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