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Chapter 1: Q. 0 (page 97)

Read the sections and make your own summary of the material.

Short Answer

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Ans: Summary of this chapter:

The infinite limitlimx→c f(x)=∞ means that for all M > 0, there exists δ > 0 such that

if x∈(c−δ,c)∪(c,c+δ),thenf(x)∈(M,∞).

The limit at infinity limx→∞ f(x)=Lmeans that for all ∈> 0, there exists N > 0 such

that

if x∈(N,∞),thenf(x)∈(L−ϵ,L+ϵ).

The infinite limit at infinity limx→∞ f(x)=∞means that for all M > 0, there exists

N > 0 such that

if x∈(N,∞), thenf(x)∈(M,∞).

Step by step solution

01

Step 1. Given information.

given,

Read the sections and make your own summary of the material.

02

Step 2. Limits Involving Infinity: 

The infinite limitlimx→c f(x)=∞means that for all M > 0, there exists δ > 0 such that

if x∈(c−δ,c)∪(c,c+δ),thenf(x)∈(M,∞).

The limit at infinity limx→∞ f(x)=Lmeans that for all ∈> 0, there exists N > 0 such

that

if localid="1649840337791" x∈(N,∞),thenf(x)∈(L−ϵ,L+ϵ).

The infinite limit at infinity llocalid="1649840366018" limx→∞ f(x)=∞means that for all M > 0, there exists

N > 0 such that

if x∈(N,∞), thenf(x)∈(M,∞).

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Most popular questions from this chapter

Find a formula for the cost Crof producing a gourmet soup can with radius rand height 5inches, and answer the following questions:

  1. What is the radius of a can that is 5inches tall and costs 30cents to produce?
  2. Your manager wants you to produce 5-inch-tall cans that cost between 20and 40cents. Write this requirement as an absolute value inequality.
  3. What range of radii would satisfy your manager? Write an absolute value inequality whose solution set lies inside this range of radii.

Write delta-epsilon proofs for each of the limit statements in Exercises 47–60.

limx→2x2-3x+2x-2=1

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

limx→-1x.

Write delta-epsilon proofs for each of the limit statements limx→cfx=Lin Exercises 47–60.

limx→1x2-1x-1=2

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

f(x)=x+1,ifx<13x-1,if1⩽x<2x+2,ifx⩾2.
See all solutions

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