Chapter 1: Q. 57 (page 108)

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

Short Answer

Expert verified

Delta-epsilon proof is,

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

Step by step solution

Step 1. Given information

We are given,

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

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 $0 < | x - 1 | < δ$ , we also have $|\frac{x^{2} - 1}{x - 1} - 2| < ϵ$ , so

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

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

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Writing the delta-epsilon proofs

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Write delta-epsilon proofs for each of the limit statements limx→cfx=Lin Exercises 47–60.limx→1x2-1x-1=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

Decision Maths

Pure Maths

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Mechanics Maths

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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