Let f(x) =$$x^2$$−2x+3.a. For ε=0.25, find the largest value of δ\u003e0 satisfying the statement|f(x)−2|\u003cε whenever 0\u003c|x−1|\u003cδ.

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

All textbooks

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.7.50

Chapter 2, Problem 2.7.50

Let f(x) =x^2−2x+3.

a. For ε=0.25, find the largest value of δ>0 satisfying the statement

|f(x)−2|<ε whenever 0<|x−1|<δ.

Verified step by step guidance

Step 1: Start by understanding the problem. We need to find the largest value of $ \delta > 0 $ such that $ |f(x) - 2| < \varepsilon $ whenever $ 0 < |x - 1| < \delta $. Here, $ f(x) = x^2 - 2x + 3 $ and $ \varepsilon = 0.25 $.

Step 2: Substitute $ f(x) = x^2 - 2x + 3 $ into the inequality $ |f(x) - 2| < 0.25 $. This gives us $ |x^2 - 2x + 3 - 2| < 0.25 $, which simplifies to $ |x^2 - 2x + 1| < 0.25 $.

Step 3: Recognize that $ x^2 - 2x + 1 $ can be rewritten as $ (x - 1)^2 $. Therefore, the inequality becomes $ |(x - 1)^2| < 0.25 $.

Step 4: Since $ |(x - 1)^2| = (x - 1)^2 $ (as squares are always non-negative), we have $ (x - 1)^2 < 0.25 $. Take the square root of both sides to get $ |x - 1| < \sqrt{0.25} $.

Step 5: Calculate $ \sqrt{0.25} $, which is 0.5. Therefore, $ \delta $ can be any value less than or equal to 0.5. The largest value of $ \delta $ satisfying the condition is $ \delta = 0.5 $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit Definition

The limit definition in calculus describes how a function behaves as it approaches a certain point. In this context, we are interested in the behavior of f(x) as x approaches 1. The statement |f(x)−2|<ε indicates that we want the function's value to be within ε of 2, which is the limit we are examining.

Epsilon-Delta Definition of Limit

The epsilon-delta definition formalizes the concept of limits in calculus. It states that for every ε>0, there exists a δ>0 such that if |x−c|<δ, then |f(x)−L|<ε. In this problem, c is 1 and L is 2, meaning we need to find a δ that ensures the function's output remains close to 2 when x is near 1.

Quadratic Functions

Quadratic functions are polynomial functions of the form f(x) = ax^2 + bx + c. They have a parabolic shape and can be analyzed using their vertex, axis of symmetry, and roots. In this case, f(x) = x^2 - 2x + 3 is a quadratic function, and understanding its graph helps in determining how it behaves around the point x=1.

Recommended video:

6:04

Introduction to Polynomial Functions

Related Practice

Textbook Question

Determine the following limits.

lim x→1 x^3 − 7x^2 + 12x / 4 − x

views

Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→0 x^2=0 (Hint: Use the identity √x2=|x|.)

Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2−25 / x−5; a=5

views

Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {x^2+1 if x≤−1

3 if x>−1; a=−1

Textbook Question

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

views

Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→3 x − 3 /|x − 3|

views