Chapter 1: Q. 54 (page 108)

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .
$\lim_{x \to 3} (x^{2} - 6 x + 11) = 2$ .

Short Answer

Expert verified

Delta-epsilon proofs for $\lim_{x \to 3} (x^{2} - 6 x + 11) = 2$ is, whenever $0 < |x - 3| < δ$ , we also havelocalid="1648023094044" $|(x^{2} - 6 x + 11) - 2| < ε$ .

Step by step solution

Step 1. Given information

$\lim_{x \to 3} (x^{2} - 6 x + 11) = 2$ .

Step 2. Consider ε>0, choose δ=ε.

For all $x$ with $0 < |x - 3| < δ$ , we also have localid="1648023140167" $|(x^{2} - 6 x + 11) - 2| < ε$ .

localid="1648023199728" $|(x^{2} - 6 x + 11) - 2| = |x^{2} - 6 x + 11 - 2| = |x^{2} - 6 x + 9| = |{(x - 3)}^{2}| = {|x - 3|}^{2} < {(δ)}^{2} = {(\sqrt{ε})}^{2} = ε$

Therefore, whenever $0 < |x - 3| < δ$ , we also havelocalid="1648023210172" $|(x^{2} - 6 x + 11) - 2| < ε$ .

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Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .
$\lim_{x \to 3} (x^{2} - 6 x + 11) = 2$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider ε>0, choose δ=ε.

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Write delta-epsilon proofs for each of the limit statements limx→cfx=Lin Exercises 47-60.limx→3x2-6x+11=2.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider ε>0, choose δ=ε.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

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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 $47 - 60$ .
$\lim_{x \to 3} (x^{2} - 6 x + 11) = 2$ .