Short Answer

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

Delta-epsilon proof is,

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

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TABLE OF CONTENTS

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 $|\frac{x^{2} - 3 x + 2}{x - 2} - 1| < ϵ$ .

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Calculus

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Short Answer

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

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Step 1. Given information

Step 2. Writing the delta-epsilon proofs

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Calculus

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

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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 in Exercises $47 - 60 .$
$\lim_{x \to 2} \frac{x^{2} - 3 x + 2}{x - 2} = 1$