Chapter 1: Q. 82 (page 122)

Write a delta–epsilon proof that shows that the function $f (x) = 2 x + 1$ is continuous at $x = 5$ . (This exercise depends on Section 1.3.)

Short Answer

Expert verified

Hence we proved $\lim_{x \to 5} (2 x + 1) = 11$ .

Step by step solution

Step 1. Given Information

We are given the function $f (x) = 2 x + 1$ is continuous at $x = 5$ and we need to write a delta–epsilon proof.

Step 2. Delta–epsilon proof

$f (x) = 2 x + 1 c = 5 L = 2 (5) + 1 = 10 + 1 = 11$

For $ε > 0, δ = \frac{ε}{2}$ ,

If $0 < |x - 5| < δ$ we have,

$|(2 x + 1) - 11| = |2 x - 10| = |2 (x - 5)| < 2 δ = 2 (\frac{ε}{2}) = ε ∴ \lim_{x \to 5} (2 x + 1) = 11$

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.

Write a delta–epsilon proof that shows that the function $f (x) = 2 x + 1$ is continuous at $x = 5$ . (This exercise depends on Section 1.3.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Delta–epsilon proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Write a delta–epsilon proof that shows that the function f(x)=2x+1 is continuous at x=5. (This exercise depends on Section 1.3.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Delta–epsilon proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Write a delta–epsilon proof that shows that the function $f (x) = 2 x + 1$ is continuous at $x = 5$ . (This exercise depends on Section 1.3.)