/*! 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 9 In terms of limits, what does it... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 9

In terms of limits, what does it mean for \(f\) to grow faster than \(g\) as \(x \rightarrow \infty ?\)

Short Answer

Expert verified
Answer: For a function \(f(x)\) to grow faster than a function \(g(x)\) as \(x\) approaches infinity, the limit of the ratio \(\frac{f(x)}{g(x)}\) as \(x\) approaches infinity must be infinity, i.e., \(\lim_{x\to\infty} \frac{f(x)}{g(x)} = \infty\).

Step by step solution

01

Set up the problem

Given functions \(f(x)\) and \(g(x)\), we want to know what it means for \(f\) to grow faster than \(g\) as \(x \rightarrow \infty\). We will examine the ratio \(\frac{f(x)}{g(x)}\) and find its limit as \(x\) approaches infinity.
02

Determine the growth factor

To determine if \(f\) grows faster than \(g\), we look at the limit of the ratio \(\frac{f(x)}{g(x)}\) as \(x\) approaches infinity. Mathematically, we write this as: $$\lim_{x\to\infty} \frac{f(x)}{g(x)}$$
03

Check the limit

If the limit of the ratio is infinity, that means \(f(x)\) grows faster than \(g(x)\) as \(x\to\infty\). In mathematical terms, we would say: $$\lim_{x\to\infty} \frac{f(x)}{g(x)} = \infty$$ If the limit has other values or does not exist, then \(f\) does not grow faster than \(g\). In conclusion, for function \(f(x)\) to grow faster than function \(g(x)\) as \(x \rightarrow \infty\), their ratio must satisfy the condition: $$\lim_{x\to\infty} \frac{f(x)}{g(x)} = \infty$$

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

A tangent question Verify by graphing that the graphs of \(y=\sin x\) and \(y=x / 2\) have one point of intersection, for \(x>0\) whereas the graphs of \(y=\sin x\) and \(y=x / 9\) have three points of intersection, for \(x>0 .\) Approximate the value of \(a\) such that the graphs of \(y=\sin x\) and \(y=x / a\) have exactly two points of intersection, for \(x>0\).

Suppose that object \(A\) is located at \(s=0\) at time \(t=0\) and starts moving along the \(s\) -axis with a velocity given by \(v(t)=2 a t,\) where \(a>0 .\) Object \(B\) is located at \(s=c>0\) at \(t=0\) and starts moving along the \(s\) -axis with a constant velocity given by \(V(t)=b>0 .\) Show that \(\mathrm{A}\) always overtakes B at time $$t=\frac{b+\sqrt{b^{2}+4 a c}}{2 a}$$

Find the solution of the following initial value problems. $$p^{\prime}(t)=10 e^{-t_{t}} ; p(0)=100$$

Is it possible? Determine whether the following properties can be satisfied by a function that is continuous on \((-\infty, \infty)\). If such a function is possible, provide an example or a sketch of the function. If such a function is not possible, explain why. a. A function \(f\) is concave down and positive everywhere. b. A function \(f\) is increasing and concave down everywhere. c. A function \(f\) has exactly two local extrema and three inflection points. d. A function \(f\) has exactly four zeros and two local extrema.

Concavity of parabolas Consider the general parabola described by the function \(f(x)=a x^{2}+b x+c .\) For what values of \(a, b\) and \(c\) is \(f\) concave up? For what values of \(a, b,\) and \(c\) is \(f\) concave down?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Decision 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.