/*! 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 10 In terms of limits, what does it... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 10

In terms of limits, what does it mean for the rates of growth of \(f\) and \(g\) to be comparable as \(x \rightarrow \infty ?\)

Short Answer

Expert verified
Answer: The rates of growth of functions f and g are considered comparable as x approaches infinity if there exists a nonzero constant k such that the limit of the ratio of f(x) to g(x) or g(x) to f(x) is equal to k: \(\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = k \neq 0\) or \(\lim_{x \rightarrow \infty} \frac{g(x)}{f(x)} = k \neq 0\)

Step by step solution

01

Define the growth rates of f and g

We begin by considering the growth rates of the functions f(x) and g(x) as x approaches infinity. The growth rate of the function will tell us how fast the function is increasing or decreasing as x increases. In this context, growth rate refers to the limit of f(x) and g(x) as x goes to infinity.
02

Introduce the concept of comparable growth rates

Two functions have comparable growth rates if their limits, as x goes to infinity, are related by a nonzero constant. In other words, the functions f and g are said to have comparable growth rates if there exists a constant k 鈮� 0 such that one function is equal to the other function times the constant k when x approaches infinity. Mathematically, this can be expressed as: \(\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = k \neq 0\) or \(\lim_{x \rightarrow \infty} \frac{g(x)}{f(x)} = k \neq 0\)
03

Provide the definition for comparable growth rates in terms of limits

So, in terms of limits, the rates of growth of the functions f and g are said to be comparable as x approaches infinity if and only if: \(\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = k \neq 0\) or \(\lim_{x \rightarrow \infty} \frac{g(x)}{f(x)} = k \neq 0\) If this condition is satisfied, we can say that the growth rates of f and g are comparable as x goes to infinity.

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!

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 functions \(f(x)=a x^{2},\) where \(a>0\) are concave up for all \(x\). Graph these functions for \(a=1,5,\) and 10, and discuss how the concavity varies with \(a\). How does \(a\) change the appearance of the graph?

Consider the quartic (fourth-degree) polynomial \(f(x)=x^{4}+b x^{2}+d\) consisting only of even-powered terms. a. Show that the graph of \(f\) is symmetric about the \(y\) -axis. b. Show that if \(b \geq 0\), then \(f\) has one critical point and no inflection points. c. Show that if \(b<0,\) then \(f\) has three critical points and two inflection points. Find the critical points and inflection points, and show that they alternate along the \(x\) -axis. Explain why one critical point is always \(x=0\) d. Prove that the number of distinct real roots of \(f\) depends on the values of the coefficients \(b\) and \(d,\) as shown in the figure. The curve that divides the plane is the parabola \(d=b^{2} / 4\) e. Find the number of real roots when \(b=0\) or \(d=0\) or \(d=b^{2} / 4\)

Consider the functions \(f(x)=\frac{1}{x^{2 n}+1},\) where \(n\) is a positive integer. a. Show that these functions are even. b. Show that the graphs of these functions intersect at the points \(\left(\pm 1, \frac{1}{2}\right),\) for all positive values of \(n\) c. Show that the inflection points of these functions occur at \(x=\pm \sqrt[2 n]{\frac{2 n-1}{2 n+1}},\) for all positive values of \(n\) d. Use a graphing utility to verify your conclusions. e. Describe how the inflection points and the shape of the graphs change as \(n\) increases.

Determine whether the following statements are true and give an explanation or counterexample. a. \(F(x)=x^{3}-4 x+100\) and \(G(x)=x^{3}-4 x-100\) are antiderivatives of the same function. b. If \(F^{\prime}(x)=f(x),\) then \(f\) is an antiderivative of \(F\) c. If \(F^{\prime}(x)=f(x),\) then \(\int f(x) d x=F(x)+C\) d. \(f(x)=x^{3}+3\) and \(g(x)=x^{3}-4\) are derivatives of the same function. e. If \(F^{\prime}(x)=G^{\prime}(x),\) then \(F(x)=G(x)\)

Use the identities \(\sin ^{2} x=(1-\cos 2 x) / 2\) and \(\cos ^{2} x=(1+\cos 2 x) / 2\) to find \(\int \sin ^{2} x d x\) and \(\int \cos ^{2} x d x\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Mechanics Maths

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.