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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 2.5.56b
Chapter 2, Problem 2.5.56b

Complete the following steps for the given functions. 


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


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

Verified step by step guidance
1
Identify the vertical asymptotes by finding the values of x that make the denominator zero.
Set the denominator equal to zero: 3x + 4 = 0.
Solve the equation for x: 3x = -4.
Divide both sides by 3 to isolate x: x = -\(\frac{4}{3}\).
Conclude that the vertical asymptote is at x = -\(\frac{4}{3}\).

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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. 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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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, particularly their asymptotic behavior, is crucial for analyzing their graphs and identifying points of discontinuity.
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Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. The degree of a polynomial indicates its highest power, which influences its end behavior and the number of roots it can have. In the context of rational functions, the behavior of the numerator polynomial is essential for determining limits and asymptotic behavior.
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Related Practice
Textbook Question

Determine the following limits.


b. limx→2−1x(x−2){\(\displaystyle\]\lim\)_{x\(\to\)2^{-}}}\(\frac{1}{\sqrt{x\left(x-2\right)}\)}

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

Textbook Question

Assume you invest \(250 at the end of each year for 10 years at an annual interest rate of rr. The amount of money in your account after 10 years is given by A(r)=250((1+r)10−1)rA\left(r\right)=\frac{250\left(\left(1+r\right)^{10}-1\right)}{r}. Assume your goal is to have \)3500 in your account after 10 years.


b. Use a calculator to estimate the interest rate required to reach your financial goal.

Textbook Question

Let g(x)={x2+xif x<1aif x=13x+5if x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)

b. Determine the value of aa for which gg is continuous from the right at 11. 

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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.

Textbook Question

Complete the following sentences in terms of a limit.


b. A function is continuous from the right at a if _____ .