Chapter 1: Q. 58 (page 108)

Write delta-epsilon proofs for each of the limit statements in Exercises $47 - 60 .$
$\lim_{x \to 2} \frac{x^{2} - 3 x + 2}{x - 2} = 1$

Short Answer

Expert verified

Delta-epsilon proof is,

Whenever $0 < | x - 2 | < δ$ , we also have $|\frac{x^{2} - 3 x + 2}{x - 2} - 1| < ϵ$ .

Step by step solution

Step 1. Given information

We are given,

$\lim_{x \to 2} \frac{x^{2} - 3 x + 2}{x - 2} = 1$

Step 2. Writing the delta-epsilon proofs

The strategy is to write delta-epsilon proofs for the given limit statement.

Consider that $\in > 0$ , choose $δ = ϵ$ .

For all x with $0 < | x - 2 | < δ$ , we also have $|\frac{x^{2} - 3 x + 2}{x - 2} - 1| < ϵ$ .

$|\frac{x^{2} - 3 x + 2}{x - 2} - 1| = |\frac{x^{2} - 3 x + 2 - 1 (x - 2)}{x - 2}| = |\frac{x^{2} - 3 x + 2 - x + 2}{x - 2}| = |\frac{x^{2} - 4 x + 4}{x - 2}| = |\frac{(x - 2)^{2}}{x - 2}| = | x - 2 | < δ = \in$

So, whenever $0 < | x - 2 | < δ$ , we also have $|\frac{x^{2} - 3 x + 2}{x - 2} - 1| < ϵ$ .

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.

Write delta-epsilon proofs for each of the limit statements in Exercises $47 - 60 .$
$\lim_{x \to 2} \frac{x^{2} - 3 x + 2}{x - 2} = 1$

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

Theoretical and Mathematical Physics

Geometry

Applied Mathematics

Calculus

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Write delta-epsilon proofs for each of the limit statements in Exercises 47–60.limx→2x2-3x+2x-2=1

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

Theoretical and Mathematical Physics

Geometry

Applied Mathematics

Calculus

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Write delta-epsilon proofs for each of the limit statements in Exercises $47 - 60 .$
$\lim_{x \to 2} \frac{x^{2} - 3 x + 2}{x - 2} = 1$