Chapter 1: Q. 44 (page 97)

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 $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
$\lim_{x \to 2} x^{3} = 8, ε = 0.25$

Short Answer

Expert verified

The largest value of $δ \approx 0.0206$ .

Step by step solution

Step 1. Given Information.

The given expression is $\lim_{x \to 2} x^{3} = 8, ε = 0.25$ .

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

Now, the largest value of delta is given by,

$x^{3} = L + ε x^{3} = 8 + 0.25 x^{3} = 8.25 x = {(8.25)}^{\frac{1}{3}}$

Thus,

$δ = {(8.25)}^{\frac{1}{3}} - 2 δ \approx 0.0206$

Step 3. Conclusion.

The largest value of $δ \approx 0.0206$ .

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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 $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
$\lim_{x \to 2} x^{3} = 8, ε = 0.25$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Conclusion.

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For each limit limx→cf(x)=Lin Exercises 43–54, use graphs and algebra to approximate the largest value of δsuch that if x∈(c-δ,c)∪(c,c+δ)thenf(x)∈(L-ε,L+ε).limx→2x3=8,ε=0.25

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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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 $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
$\lim_{x \to 2} x^{3} = 8, ε = 0.25$