Chapter 1: Q. 64 (page 108)

For each of the limit statements in Exercises 61-66, write a $δ - M, N - ϵ$ , or $N - M$ proof, according to the type of limit statement.
$\lim_{x \to - \infty} \frac{2 x - 1}{x} = 2$

Short Answer

Expert verified

The proof of given limit $\lim_{x \to - \infty} \frac{2 x - 1}{x} = 2$ is,

$N = \frac{1}{ε}$ .

Step by step solution

Step 1. Given information

We are given,

$\lim_{x \to - \infty} \frac{2 x - 1}{x} = 2$

Step 2. Writing the delta-epsilon proofs

The strategy is to use algebra to find N in terms of $ε$ ,

From the given limit, we get

$f (x) = \frac{2 x - 1}{x} c = - \infty L = 2$

Given $ε < 0$ , choose $N = \frac{1}{ε}$ .

If $x < N$ , we have,

$|\frac{2 x - 1}{x} - 2| = |\frac{2 x - 1 - 2 x}{x}| = |\frac{- 1}{x}| = |\frac{1}{x}| < \frac{1}{N} = ε$

Hence, $N = \frac{1}{ε}$ .

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

For each of the limit statements in Exercises 61-66, write a $δ - M, N - ϵ$ , or $N - M$ proof, according to the type of limit statement.
$\lim_{x \to - \infty} \frac{2 x - 1}{x} = 2$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Writing the delta-epsilon proofs

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Applied Mathematics

Statistics

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

For each of the limit statements in Exercises 61-66, write a δ-M,N-ϵ, or N-M proof, according to the type of limit statement.limx→-∞2x-1x=2

Short Answer

Step by step solution

Step 1. Given information

Step 2. Writing the delta-epsilon proofs

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Applied Mathematics

Statistics

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

For each of the limit statements in Exercises 61-66, write a $δ - M, N - ϵ$ , or $N - M$ proof, according to the type of limit statement.
$\lim_{x \to - \infty} \frac{2 x - 1}{x} = 2$