Chapter 1: Q. 63 (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,

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

Hence, $N = \frac{1}{ε}$

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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

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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

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