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 2.6

Chapter 2, Problem 2.6

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>

Verified step by step guidance

Step 1: Identify the function f(x) and analyze its form. If the function is not explicitly given, assume it is a piecewise function or a rational function, as these often have discontinuities.

Step 2: Determine the types of discontinuities that can occur. Common types include removable discontinuities (holes), jump discontinuities, and infinite discontinuities (vertical asymptotes).

Step 3: Check for points where the function is undefined within the interval (0, 5). For rational functions, this occurs where the denominator is zero. For piecewise functions, check the transition points.

Step 4: Apply the continuity checklist at each point of interest: (a) The function must be defined at the point, (b) The limit of the function as it approaches the point from both sides must exist, and (c) The limit must equal the function's value at that point.

Step 5: For each point of discontinuity identified, specify which condition(s) from the continuity checklist are violated. This will help in classifying the type of discontinuity present.

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Key Concepts

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

Continuity of Functions

A function is continuous at a point if three conditions are met: the function is defined at that point, the limit of the function as it approaches that point exists, and the limit equals the function's value at that point. Understanding these conditions is crucial for identifying points of discontinuity.

Types of Discontinuities

Discontinuities can be classified into three main types: removable, jump, and infinite. A removable discontinuity occurs when a function can be made continuous by redefining a point, a jump discontinuity involves a sudden change in function value, and an infinite discontinuity occurs when the function approaches infinity at a point.

The Continuity Checklist

The continuity checklist is a systematic approach to determine if a function is continuous at a point. It includes checking if the function is defined at the point, if the limit exists as the input approaches the point, and if the limit equals the function's value. Violations of any of these conditions indicate a discontinuity.

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Related Practice

Textbook Question

b. Estimate a solution to the equation in the given interval using a root finder.

x^5+7x+5=0; (−1,0)

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Textbook Question

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist.

{2,3/4,4/9,5/16,…}, which is defined by f(n) = (n+1) / n^2, for n=1,2,3,…

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Textbook Question

Use a graph of f to estimate $\lim_{x \to a} f (x)$ or to show that the limit does not exist. Evaluate f(x) near $x = a$ to support your conjecture.

$f (x) = \frac{1 - \cos (2 x - 2)}{{(x - 1)}^{2}}; a = 1$

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Textbook Question

Consider the position function s(t)=−16t^2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=1. <IMAGE>

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Textbook Question

The following table gives the position $s$\left$(t$\right$)$ of an object moving along a line at time $t$ . Determine the average velocities over the time intervals $\left\[\lbrack\)1,1.01$\right\]\rbrack$ , $\left\[\lbrack$1,1.001$\right\]\rbrack$ , and $\(\left\]\lbrack$1,1.0001^{}$\right$.]$ . Then make a conjecture about the value of the instantaneous velocity at $t=1$ . <IMAGE>

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Textbook Question

Suppose $\lim_{x \to a} f (x) = L$ and $\lim_{x \to a} g (x) = M$ . Prove that $\lim_{x \to a} (f (x) - g (x)) = L - M$ .

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