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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 2.7.49
Chapter 2, Problem 2.7.49

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume lim x→a f(x) =L


d. If |x−a|<δ, then a−δ<x<a+δ.

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1
Step 1: Understand the statement: The statement is about the definition of the limit of a function as x approaches a. It involves the concept of a delta (δ) neighborhood around the point a.
Step 2: Recall the definition of a limit: The limit of f(x) as x approaches a is L, written as \( \lim_{{x \to a}} f(x) = L \), if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).
Step 3: Analyze the given condition: The condition \( |x - a| < \delta \) implies that x is within a distance δ from a. This means x is in the interval (a - δ, a + δ).
Step 4: Verify the statement: The statement 'If \( |x - a| < \delta \), then \( a - \delta < x < a + \delta \)' is indeed true by the definition of absolute value and the concept of a δ-neighborhood.
Step 5: Conclusion: The statement is true because it directly follows from the definition of the absolute value and the concept of a δ-neighborhood around a point a.

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Key Concepts

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

Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a certain value. In this case, lim x→a f(x) = L means that as x gets closer to a, the function f(x) approaches the value L. Understanding limits is crucial for analyzing continuity and the behavior of functions near specific points.
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Delta-Epsilon Definition of a Limit

The delta-epsilon definition formalizes the concept of limits. It states that for every ε (epsilon) greater than 0, there exists a δ (delta) such that if |x - a| < δ, then |f(x) - L| < ε. This definition is essential for proving the validity of limits and understanding the precise conditions under which a function approaches a limit.
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Inequalities and Intervals

Inequalities are used to describe the range of values that a variable can take. The statement |x - a| < δ implies that x lies within the interval (a - δ, a + δ). Understanding how to manipulate and interpret these inequalities is vital for analyzing the behavior of functions in the context of limits and continuity.
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Related Practice
Textbook Question

Determine the following limits.


limz→4z−5(z2−10z+24)2{\(\displaystyle\]\lim\)_{z\(\to\)4}}\(\frac{z-5}{\left(z^2-10z+24\right)^2}\)

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Textbook Question

Determine the points on the interval (0, 5) at which the following functions f have discontinuities. At each point of discontinuity, state the conditions in the continuity checklist that are violated. <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→2 (x^2+3x)=10

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Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=4x(3x−9x2+1)f\(\left\)(x\(\right\))=4x\(\left\)(3x-\(\sqrt{9x^2+1}\]\right\))

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Textbook Question

Determine the following limits. 

lim x→−∞ (3x7 + x2) 

Textbook Question

Determine the following limits.


lim x→π/2 1/√sin x − 1 / x + π/2