Short Answer

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .
$\lim_{x \to 2} (3 - 4 x) = - 5$ .

Delta-epsilon proof for $\lim_{x \to 2} (3 - 4 x) = - 5$ is, whenever localid="1648006806212" $0 < |x - 2| < δ$ , we also have localid="1648021669064" $|(3 - 4 x) + 5| < ε$ .

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Step by Step Solution

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Step 1. Given information

$\lim_{x \to 2} (3 - 4 x) = - 5$ .

Step 2. Consider ε>0, choose δ=ε4.

For all $x$ with localid="1648006934128" $0 < |x - 2| < δ$ ,we also have localid="1648021795730" $|(3 - 4 x) + 5| < ε$ .

localid="1648021807616" $|(3 - 4 x) + 5| = |3 - 4 x + 5| = |8 - 4 x| = 4 |2 - x| = 4 |x - 2| < 4 δ = 4 (\frac{ε}{4}) = ε$

Therefore, whenever $0 < |x - 2| < δ$ , we also have localid="1648021825608" $|(3 - 4 x) + 5| < ε$ .

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Calculus

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Short Answer

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .
$\lim_{x \to 2} (3 - 4 x) = - 5$ .

Step by Step Solution

Step 1. Given information

Step 2. Consider ε>0, choose δ=ε4.

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Write delta-epsilon proofs for each of the limit statements limx→cfx=L in Exercises 47-60.limx→23-4x=-5.

Step by Step Solution

Step 1. Given information

Step 2. Consider ε>0, choose δ=ε4.

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Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .
$\lim_{x \to 2} (3 - 4 x) = - 5$ .