/*! 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 Evaluate \(\lim _{n \rightarrow ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 9

Evaluate \(\lim _{n \rightarrow \infty} n\left(\ln \left(3+\frac{1}{n}\right)-\ln 3\right)\).

Short Answer

Expert verified
The limit of the given expression is \(-\frac{1}{3}\).

Step by step solution

01

Rewrite the given expression using logarithm properties

Recall that the subtraction of logarithms with the same base can be expressed as a single logarithm whose argument is the ratio of the arguments of the original logarithms. Rewrite the expression using the logarithm properties: \[\lim_{n\to\infty} n\left(\ln\left(3+\frac{1}{n}\right)-\ln 3\right) = \lim_{n\to\infty} n\ln\left(\frac{3+\frac{1}{n}}{3}\right)\]
02

Simplify the expression inside the logarithm

Simplify the expression inside the logarithm by combining the fractions: \[\lim_{n\to\infty} n\ln\left(\frac{3+\frac{1}{n}}{3}\right) = \lim_{n\to\infty} n\ln\left(\frac{3n+1}{3n}\right)\]
03

Rewrite the expression as a quotient

Recall that L'Hopital's rule requires that the limit be in the form \(\lim_{x\to a} \frac{f(x)}{g(x)}\). Rewrite the expression as a quotient using the property that the logarithm of a number raised to a power is equal to the power times the logarithm: \[\lim_{n\to\infty} n\ln\left(\frac{3n+1}{3n}\right) = \lim_{n\to\infty} \frac{\ln\left(\frac{3n+1}{3n}\right)}{\frac{1}{n}}\]
04

Apply L'Hopital's rule

Now that the expression is in the form required to apply L'Hopital's rule, differentiate both the numerator and the denominator with respect to \(n\): \[\lim_{n\to\infty} \frac{\ln\left(\frac{3n+1}{3n}\right)}{\frac{1}{n}} = \lim_{n\to\infty} \frac{\frac{d}{dn}\ln\left(\frac{3n+1}{3n}\right)}{\frac{d}{dn}\frac{1}{n}}\] Using the chain rule, we find the derivative of the numerator: \[\frac{d}{dn}\ln\left(\frac{3n+1}{3n}\right) = \frac{1}{\frac{3n+1}{3n}} \cdot \frac{d}{dn}\left(\frac{3n+1}{3n}\right) = \frac{3n}{3n+1} \cdot \frac{-3}{9n^2}\] And the derivative of the denominator: \[\frac{d}{dn} \frac{1}{n} = -\frac{1}{n^2}\]
05

Simplify the expression and take the limit

Now substitute the derivatives back into the limit: \[\lim_{n\to\infty} \frac{\frac{3n}{3n+1} \cdot \frac{-3}{9n^2}}{-\frac{1}{n^2}}\] Simplify the expression: \[\lim_{n\to\infty} \frac{3n}{3n+1} \cdot \frac{-3n^2}{9n^2} = \lim_{n\to\infty} \left(\frac{3n}{3n+1}\right) \left(\frac{-3}{9}\right)\] Further simplify: \[\lim_{n\to\infty} \left(\frac{3n}{3n+1}\right) \left(-\frac{1}{3}\right) = -\frac{1}{3} \lim_{n\to\infty} \left(\frac{3n}{3n+1}\right)\] Now take the limit as \(n\) approaches infinity: \[-\frac{1}{3} \lim_{n\to\infty} \left(\frac{3n}{3n+1}\right) = -\frac{1}{3} \cdot \frac{\infty}{\infty+1} = -\frac{1}{3}\] So, the limit of the given expression is \(-\frac{1}{3}\).

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!

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

Evaluate \(\lim _{n \rightarrow \infty}\left(1+\frac{3}{n}\right)^{n}\).

Use Pascal's triangle to simplify the indicated expression. $$ (3-\sqrt{2})^{6} $$

Evaluate the geometric series. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots+\frac{1}{2^{80}}-\frac{1}{2^{81}} $$

Evaluate the arithmetic series. $$ 1+2+3+\cdots+98+99+100 $$

Evaluate \(\lim _{n \rightarrow \infty} n\left(e^{1 / n}-1\right)\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

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.