/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 293 Graph the function on a graphing... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 293

Graph the function on a graphing calculator on the window \(x=[-5,5]\) and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit. \(\lim _{x \rightarrow \infty} \frac{3 x+2}{x+5}\)

Short Answer

Expert verified
The horizontal asymptote of \( \frac{3x+2}{x+5} \) is \( y=3 \).

Step by step solution

01

Understanding the Function

The given function is \( f(x) = \frac{3x + 2}{x + 5} \). Here, we are asked to graph this function using a graphing calculator and find the horizontal asymptote by estimating and calculating it.
02

Graph the Function

Using a graphing calculator, plot the function \( f(x) = \frac{3x + 2}{x + 5} \) with the window setting for \( x \) from -5 to 5. Identify the horizontal direction in which the function seems to approach a constant value.
03

Estimate the Horizontal Asymptote

Observe the graph you produced. As \( x \) approaches both positive and large values, the function approaches a particular y-value. Estimate what this y-value appears to be from the graph.
04

Set Up the Limit Calculation

The formal way to find the horizontal asymptote is through limit calculation: \( \lim_{{x \to \infty}} \frac{3x + 2}{x + 5} \). Factoring \( x \) out from the numerator and the denominator gives \( \frac{3 + \frac{2}{x}}{1 + \frac{5}{x}} \).
05

Apply Limit Properties

As \( x \) approaches infinity, the terms \( \frac{2}{x} \) and \( \frac{5}{x} \) both approach 0. Thus, the expression simplifies to \( \frac{3 + 0}{1 + 0} = 3 \).
06

Conclude Horizontal Asymptote

Hence, \( \, \lim_{{x \to \infty}} \frac{3x + 2}{x + 5} = 3 \). The horizontal asymptote is the line \( y = 3 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Calculator
A graphing calculator is a powerful tool that helps visualize mathematical functions. For this exercise, it is used to graph the function \( f(x) = \frac{3x + 2}{x + 5} \) over the window \( x = [-5, 5] \). When using a graphing calculator, follow these basic steps:
  • Enter the function into the calculator's function editor.
  • Set the viewing window from \( x = -5 \) to \( x = 5 \), which lets you see how the function behaves in this region.
  • Plot the graph to observe the curve as it moves towards larger x-values.
By observing the graph, you can identify the behavior of the function over a range of values and estimate where it flattens out.
Limit Calculation
The concept of a limit is fundamental to understanding horizontal asymptotes. For the function \( f(x) = \frac{3x + 2}{x + 5} \), you want to determine its behavior as \( x \) approaches infinity. To calculate the limit, start by expressing it as:\[ \lim_{x \to \infty} \frac{3x + 2}{x + 5} \]Breaking down this expression helps you understand how the function behaves over large values of \( x \).Substitute terms in the numerator and the denominator to factor out \( x \):\[ \frac{3 + \frac{2}{x}}{1 + \frac{5}{x}} \]Notice that as \( x \) increases, \( \frac{2}{x} \) and \( \frac{5}{x} \) approach zero. This simplification gives the limit as 3.
Infinite Limit
An infinite limit refers to the behavior of a function as the input grows indefinitely large. In this context, we're looking at what happens to \( f(x) = \frac{3x + 2}{x + 5} \) as \( x \to \infty \). For rational functions such as this one, the highest degree terms in the numerator and denominator dictate the limit.By simplifying \( \frac{3x + 2}{x + 5} \) to \( \frac{3}{1} \) as \( x \to \infty \), you can see that the infinite limit results in the value 3. This calculation not only finds the limit but also confirms the horizontal asymptote at \( y = 3 \), indicating that the function approaches this y-value.
Rational Functions
Rational functions are quotients of polynomial expressions. In our case, the function \( f(x) = \frac{3x + 2}{x + 5} \) is a simple example. You can express the behavior of such functions through their limits and asymptotes.Important properties of rational functions include:
  • Vertical asymptotes occur where the denominator is zero.
  • Horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials.
For our function, because the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of their leading coefficients, which results in \( y = 3 \). This key insight helps understand and solve similar problems involving rational functions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A rocket is launched into space; its kinetic energy is given by \(K(t)=\left(\frac{1}{2}\right) m(t) v(t)^{2},\) where \(K\) is the kinetic energy in joules, \(m\) is the mass of the rocket in kilograms, and \(v\) is the velocity of the rocket in meters/second. Assume the velocity is increasing at a rate of 15 \(\mathrm{m} / \mathrm{sec}^{2}\) and the mass is decreasing at a rate of 10 kg/sec because the fuel is being burned. At what rate is the rocket's kinetic energy changing when the mass is 2000 kg and the velocity is 5000 \(\mathrm{m} / \mathrm{sec}\) ? Give your answer in mega-Joules (MI), which is equivalent to \(10^{6} \mathrm{J} .\)

Determine whether the Mean Value Theorem applies for the functions over the given interval \([a, b] .\) Justify your answer. $$ f(x)=\tan (2 \pi x) \text { over }[0,2] $$

Evaluate the following limits. $$\lim _{x \rightarrow \infty} \cos \left(\frac{1}{x}\right)$$

Evaluate the following limits. $$\lim _{x \rightarrow \infty}(3 x)^{1 / x}$$

Use the Mean Value Theorem and find all points \(0 < c < 2\) such that \(f(2)-f(0)=f^{\prime}(c)(2-0)\) $$ f(x)=\sin (\pi x) $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.