Chapter 1: Q. 55 (page 108)

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .
$\lim_{x \to 1} (2 x^{2} - 4 x + 3) = 1$ .

Short Answer

Expert verified

Delta-epsilon proof for $\lim_{x \to 1} (2 x^{2} - 4 x + 3) = 1$ is, whenever $0 < |x - 1| < δ$ , we also havelocalid="1648022777873" $|(2 x^{2} - 4 x + 3) - 1| < ε$ .

Step by step solution

Step 1. Given information

$\lim_{x \to 1} (2 x^{2} - 4 x + 3) = 1$ .

Step 2. Consider ε>0, choose δ=ε2.

For all $x$ with $0 < |x - 1| < δ$ , we also have $|(2 x^{2} - 4 x + 3) - 1| < ε$ .localid="1648023004065" $|(2 x^{2} - 4 x + 3) - 1| = |2 x^{2} - 4 x + 3 - 1| = |2 x^{2} - 4 x + 2| = 2 |x^{2} - 2 x + 1| = 2 |{(x - 1)}^{2}| = 2 {|x - 1|}^{2} < 2 {(δ)}^{2} = 2 {(\sqrt{\frac{ε}{2}})}^{2} = ε$

Therefore, whenever $0 < |x - 1| < δ$ ,we also havelocalid="1648023016351" $|(2 x^{2} - 4 x + 3) - 1| < ε$ .

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Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .
$\lim_{x \to 1} (2 x^{2} - 4 x + 3) = 1$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider ε>0, choose δ=ε2.

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Write delta-epsilon proofs for each of the limit statements limx→cfx=Lin Exercises 47-60.limx→12x2-4x+3=1.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider ε>0, choose δ=ε2.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

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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 $47 - 60$ .
$\lim_{x \to 1} (2 x^{2} - 4 x + 3) = 1$ .