Skip to main content
Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.7.1
Chapter 2, Problem 2.7.1

Suppose x lies in the interval (1, 3) with x≠2. Find the smallest positive value of δ such that the inequality 0<|x−2|<δ is true.

Verified step by step guidance
1
Understand the problem: We need to find the smallest positive value of δ such that the inequality 0 < |x−2| < δ holds true for x in the interval (1, 3) with x ≠ 2.
Consider the interval (1, 3) and identify the points where x is closest to 2. Since x ≠ 2, we need to consider the values of x approaching 2 from both sides within the interval.
Calculate the distance from 2 to the endpoints of the interval. The distance from 2 to 1 is |2 - 1| = 1, and the distance from 2 to 3 is |3 - 2| = 1.
Since x cannot be equal to 2, the smallest positive value of δ must be less than the smallest distance from 2 to the endpoints of the interval, which is 1.
Conclude that the smallest positive value of δ is slightly less than 1, ensuring that 0 < |x−2| < δ holds true for all x in the interval (1, 3) with x ≠ 2.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Absolute Value Inequality

An absolute value inequality of the form |x - a| < δ describes the distance between x and a being less than δ. In this context, it means that x is within δ units of the point a (which is 2 here). Understanding this concept is crucial for determining the range of x values that satisfy the inequality.
Recommended video:
06:37
Average Value of a Function

Intervals

An interval is a set of real numbers that lie between two endpoints. The interval (1, 3) indicates that x can take any value greater than 1 and less than 3, excluding the endpoints. This concept is important for identifying the valid values of x that can be used in the inequality.
Recommended video:
03:38
Intro to Continuity Example 1

Limit and Continuity

In calculus, limits describe the behavior of a function as it approaches a certain point. The condition x ≠ 2 implies that we are examining the behavior of x as it approaches 2 from either side, which is essential for understanding how δ can be chosen to maintain the inequality while respecting the constraints of the interval.
Recommended video:
05:34
Intro to Continuity
Related Practice
Textbook Question

Evaluate lim x→0 x + 1/ 1 −cos x.

Textbook Question

Determine the interval(s) on which the following functions are continuous. 

f(t)=t+2 / t^2−4

14
views
Textbook Question

Evaluate limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}.


f(x)=4x3+11−x3f\(\left\)(x\(\right\))=\(\frac{4x^3+1}{1-x^3}\)

1
views
Textbook Question

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


limx→01−cos(x)cos2(x)−3cos(x)+2{\(\displaystyle\]\lim\)_{x\(\to\)0}\(\frac{1-\cos\left(x\right)}{\cos^2\left(x\right)-3\cos\left(x\right)+2}\)}

2
views
Textbook Question

Evaluate lim x→∞ f(x) and lim x→−∞ f(x) sing the figure. <IMAGE>

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→3 x^3=27

6
views