Chapter 1: Q. 43 (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 $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
role="math" localid="1648027918805" $\lim_{x \to 2} x^{3} = 8, ε = 0.5$

Short Answer

Expert verified

The largest value of $δ \approx 0.0408$ .

Step by step solution

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 $δ \approx 0.0408$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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" $\lim_{x \to 2} x^{3} = 8, ε = 0.5$

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

Logic and Functions

Probability and Statistics

Pure Maths

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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).role="math" localid="1648027918805" limx→2x3=8,ε=0.5

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

Logic and Functions

Probability and Statistics

Pure Maths

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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" $\lim_{x \to 2} x^{3} = 8, ε = 0.5$