Q. 55 Write delta-epsilon proofs for e... [FREE SOLUTION]

Chapter 1: Q. 55 (page 108)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information

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

Step 2. Consider ε>0, choose δ=ε2.

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

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

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91影视

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider ε>0, choose δ=ε2.

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91影视

Write delta-epsilon proofs for each of the limit statements limx鈫�cfx=Lin Exercises 47-60.limx鈫�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

Mechanics Maths

Pure Maths

Decision Maths

Calculus

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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