/*! 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 2 Show that \(\lim \left(n x /\lef... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 2

Show that \(\lim \left(n x /\left(1+n^{2} x^{2}\right)\right)=0\) for all \(x \in \mathbb{R}\).

Short Answer

Expert verified
The limit \(\lim \left(n x /\left(1+n^{2} x^{2}\right)\right)=0\) for all \(x \in \mathbb{R}\).

Step by step solution

01

Analyzing the Function

First, look at the function given: \(f(n, x) = n x /(1+n^{2} x^{2})\). This function is defined for any real number \(x\) and for \(n > 0\). Notice also that the function is undefined at \(x = 0\).
02

Simplifying the Expression

Next, simplify the expression in the limit. Notice that both the numerator and denominator are multiplied by \(n\), a constant. Divide numerator and denominator by \(n\). The numerator becomes \(x\) and the denominator becomes \((1/n + n x^{2})\), resulting in the function \(g(n, x) = x /(1/n + n x^{2})\).
03

Applying Limit Properties

Now, apply limit properties to the simplified function. As \(n \rightarrow \infty\), \((1/n) \rightarrow 0\) and \((n x^{2})\) remains unchanged. Therefore, the limit of the denominator as \(n\) approaches infinity becomes \((0 + n x^{2}) = n x^{2}\). The limit of the function \(g(n, x)\) as \(n \rightarrow \infty\) is then \(\lim_{n \to \infty} x /(n x^{2})\). Simplifying this by dividing the numerator and denominator by \(n\), the function becomes \(x / (nx)\). This simplifies to \(1/n\). The limit of \(1/n\) as \(n\) approaches infinity is 0.
04

Concluding the Proof

Finally, conclude the proof. Based on the analysis, the limit of the function \(\lim \left(n x /\left(1+n^{2} x^{2}\right)\right)\) as \(n\) approaches infinity for all \(x \in \mathbb{R}\) is 0. Therefore, the proof is complete.

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

If \(a>0, a \neq 1\), show that the function \(x \mapsto \log _{a} x\) is differentiable on \((0, \infty)\) and that \(D \log _{n} x=1 /(x \ln a)\) for \(x \in(0, \infty)\)

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be uniformly continuous on \(\mathbb{R}\) and let \(f_{n}(x):=f(x+1 / n)\) for \(x \in \mathbb{R}\). Show that \(\left(f_{n}\right)\) converges uniformly on \(\mathbb{R}\) to \(f\).

Show that \(\lim (\operatorname{Arctan} n x)=(\pi / 2) \operatorname{sgn} x\) for \(x \in \mathbb{R}\)

Show that the sequence \(\left(x^{2} e^{-n x}\right)\) converges uniformly on \([0, \infty)\).

Show that the sequence \(\left(\left(x^{n} /\left(1+x^{n}\right)\right)\right.\) does not converge uniformly on \(\\{0,2\\}\) by showing that the limit function is not continuous on \([0,2]\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

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.