/*! 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 12 Find the limit. $$\lim _{r \ri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 12

Find the limit. $$\lim _{r \rightarrow \infty} \frac{4 r^{3}-r^{2}}{(r+1)^{3}}$$

Short Answer

Expert verified
The limit is 4.

Step by step solution

01

Understanding the Problem

We are tasked to find the limit as \( r \) approaches infinity of the expression \( \frac{4r^{3} - r^{2}}{(r+1)^3} \). The strategy is typically to simplify the expression to see its behavior at large \( r \).
02

Simplifying the Expression

First, expand the denominator. \( (r+1)^{3} = r^{3} + 3r^{2} + 3r + 1 \). Our expression becomes: \( \frac{4r^{3} - r^{2}}{r^{3} + 3r^{2} + 3r + 1} \).
03

Finding the Dominant Terms

Identify the dominant terms in the numerator and the denominator. In the numerator, \( 4r^3 \) dominates \( -r^2 \), and in the denominator, \( r^3 \) is the dominant term.
04

Simplifying the Limit Expression

Since \( r^3 \) is the dominant term in both the numerator and the denominator, divide each term by \( r^3 \) to simplify the expression: \( \lim _{r \rightarrow \infty} \frac{4 - \frac{1}{r}}{1 + \frac{3}{r} + \frac{3}{r^2} + \frac{1}{r^3}} \).
05

Evaluating the Limit

As \( r \) approaches infinity, terms involving \( \frac{1}{r} \), \( \frac{1}{r^2} \), and \( \frac{1}{r^3} \) approach zero. Thus, the expression simplifies to \( \frac{4 - 0}{1 + 0 + 0 + 0} = 4 \).

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.

Dominant Terms
In calculus, when working with limits, identifying dominant terms is a crucial step. These are the terms that have the greatest impact on the expression as the variable, in this case, \( r \), approaches infinity.
  • Dominant terms help us focus on the parts of the expression that truly affect the limit's behavior.
  • For polynomial expressions like \( 4r^3 - r^2 \), the term with the highest degree, here \( 4r^3 \), will dominate.
  • Similarly, in the denominator \((r+1)^3\), expanding gives \( r^3 + 3r^2 + 3r + 1 \), where \( r^3 \) is the dominant term.
By concentrating on these terms, you can often simplify the calculation of limits for functions as the variable approaches infinity.
Simplifying Expressions
Simplifying expressions before evaluating a limit can make the process significantly easier. You'll want to break down complex expressions into simpler forms.
  • Begin by expanding any parts of the expression that are not in a simple polynomial form, such as \((r+1)^3\), which expands to \( r^3 + 3r^2 + 3r + 1 \).
  • Once expanded, compare the degree of terms across the numerator and the denominator to determine what can be simplified.
  • Identify common factors and divide them out. In our case, dividing all terms by \( r^3 \) works well since it鈥檚 the dominant term.
This results in a much friendlier form for limit evaluation: \( \frac{4 - \frac{1}{r}}{1 + \frac{3}{r} + \frac{3}{r^2} + \frac{1}{r^3}} \). Notice how each term in this fraction is simplified, leading to easier limit evaluation.
Limit Evaluation
The final step is to evaluate the limit, especially when approaching infinity. After simplifying the expression, this step becomes straightforward.
  • Consider the expression \( \lim_{r \to \infty} \frac{4 - \frac{1}{r}}{1 + \frac{3}{r} + \frac{3}{r^2} + \frac{1}{r^3}} \).
  • As \( r \to \infty \), terms like \( \frac{1}{r} \), \( \frac{1}{r^2} \), and \( \frac{1}{r^3} \) approach zero because a very large denominator makes the fraction very small.
  • This simplifies our expression to \( \frac{4 - 0}{1 + 0 + 0 + 0} = 4 \).
With this knowledge, it becomes clear why the limit evaluates to \( 4 \). The procedure highlights the importance of focusing on dominant terms, simplifying expressions, and understanding how these lead to the final limit outcome.

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

The given limit represents the derivative of a function \(f\) at a number \(a\). Find \(f\) and \(a\) $$\lim _{t \rightarrow 1} \frac{\sqrt{t+1}-\sqrt{2}}{t-1}$$

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=\sqrt{x+3}, \quad \text { at }(1,2)$$

Heart Rate \(A\) cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after \(t\) min. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. $$\begin{array}{|c|c|}\hline t \text { (min) } & \text { Heartbeats } \\\\\hline 36 & 2530 \\\38 & 2661 \\\40 & 2806 \\\42 & 2948 \\\44 & 3080 \\\\\hline\end{array}$$ (a) Find the average heart rates (slopes of the secant lines) over the time intervals \([40,42]\) and \([42,44]\) (b) Estimate the patient's heart rate after 42 min by averaging the slopes of these two secant lines.

The given limit represents the derivative of a function \(f\) at a number \(a\). Find \(f\) and \(a\) $$\lim _{x \rightarrow 5} \frac{2^{x}-32}{x-5}$$

Estimating Limits Numerically and Graphically Estimate the value of the limit by making a table of values. Check your work with a graph. $$\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.