/*! 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 113 Let \(a\) and \(b\) be positive ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 113

Let \(a\) and \(b\) be positive real numbers. Evaluate \(\lim _{x \rightarrow \infty}(a x-\sqrt{a^{2} x^{2}-b x})\) in terms of \(a\) and \(b\)

Short Answer

Expert verified
Answer: The limit is 0.

Step by step solution

01

Substitute

To simplify the expression, let's substitute \(u = ax\). So, the expression becomes \(u-\sqrt{u^2-b(u/a)}\). Now, we need to find the limit as \(u\rightarrow\infty\).
02

Factor out u from the square root

Let's factor out \(u\) from the square root: \(u-\sqrt{u^2-\frac{b}{a}u}\). Now notice that we can rewrite it as \(u\left(1-\sqrt{1-\frac{b}{au}}\right)\).
03

Apply limit properties

Now we need to find \(\lim_{u\rightarrow\infty}\left(1-\sqrt{1-\frac{b}{au}}\right)\). We can rewrite the expression as the product of two limits: \(\lim_{u\rightarrow\infty} u \cdot \lim_{u\rightarrow\infty}\left(1-\sqrt{1-\frac{b}{au}}\right)\).
04

Evaluate the second limit

To evaluate the second limit, notice that as \(u\rightarrow\infty\), the term inside the square root approaches 1: \(\lim_{u\rightarrow\infty}\sqrt{1-\frac{b}{au}} = \lim_{u\rightarrow\infty}\sqrt{1-0}=\lim_{u\rightarrow\infty}1\). Thus, the second limit becomes \(\lim_{u\rightarrow\infty}\left(1-\sqrt{1-\frac{b}{au}}\right) = \lim_{u\rightarrow\infty}\left(1-1\right) = 0\).
05

Evaluate the product of limits

Now we have \(\lim_{u\rightarrow\infty} u \cdot \lim_{u\rightarrow\infty}\left(1-\sqrt{1-\frac{b}{au}}\right) = \lim_{u\rightarrow\infty} u \cdot 0 = 0\). This means that \(\lim_{x\rightarrow\infty}(ax-\sqrt{a^2x^2-bx})=0\) in terms of \(a\) and \(b\).

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

Let \(a\) and \(b\) be positive real numbers. Evaluate \(\lim _{x \rightarrow \infty}(a x-\sqrt{a^{2} x^{2}-b x})\) in terms of \(a\) and \(b\)

Find the solution of the following initial value problems. $$v^{\prime}(x)=4 x^{1 / 3}+2 x^{-1 / 3} ; v(8)=40, x>0$$

Increasing and decreasing functions Find the intervals on which \(f\) is increasing and the intervals on which it is decreasing. $$f(x)=-2 x^{4}+x^{2}+10$$

Find the solution of the following initial value problems. \(y^{\prime}(\theta)=\frac{\sqrt{2} \cos ^{3} \theta+1}{\cos ^{2} \theta} ; y\left(\frac{\pi}{4}\right)=3,-\pi / 2<\theta<\pi / 2\)

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=2 e^{t}-12 ; v(0)=1, s(0)=0$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure 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.