Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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

Briggs 3rd Edition

Ch. 2 - Limits

Problem 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.

Verified step by step guidance

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.

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.

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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Textbook Question

Suppose ${$\displaystyle\]\lim$_{x$\to$ a}}$\text{ }$f$\left$(x$\right$)=L$ . Prove that $\lim_{x \to a} (c (f (x)) = c L$ , where $c$ is a constant.

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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>

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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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Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=1/ √x sec x

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