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Q. 43

Expert-verified

Calculus

Found in: Page 97

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

For each limit $\underset{x \to c}{l i m} f (x) = L$ in Exercises 43–54, use graphs and algebra to approximate the largest value of $δ$ such that if $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
role="math" localid="1648027918805" $\lim_{x \to 2} x^{3} = 8, ε = 0.5$

The largest value of $δ \approx 0.0408$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information.

The given expression is $\lim_{x \to 2} x^{3} = 8, ε = 0.5$ .

Step 2. Explanation.

From the given expression, we have, $c = 2, L = 8$ .

The limit expression can be written as a formal statement as below,

For all epsilon positive, there exists a delta positive such that if

localid="1648188825671" $x \in (2 - δ, 2) \cup (2, 2 + δ) Then x^{3} \in (8 - ε, 8 + ε)$

Now, the largest value of delta is given by,

$x^{3} = L + ε x^{3} = 8 + 0.5 x^{3} = 8.5 x = {(8.5)}^{\frac{1}{3}}$

Thus,

$δ = {(8.5)}^{\frac{1}{3}} - 2 δ \approx 0.0408$

Step 3. Conclusion.

The largest value of $δ \approx 0.0408$ .

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