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Chapter 1: Q. 60 (page 108)

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

limx鈫�2+32x-4=0

Short Answer

Expert verified

Delta-epsilon proof is,

Whenever x鈭�(2,2+未), we also have |32x-4-0|<系.

Step by step solution

01

Step 1. Given information  

We are given,

limx鈫�2+32x-4=0

02

Step 2. Writing the delta-epsilon proofs 

The strategy is to write delta-epsilon proofs for the given limit statement.

Consider that 鈭�>0, choose 未=系218.

The limit statement limx鈫�2+32x-4=0means that for all 鈭�>0, there exist 未>0such that if x鈭�(2,2+未), then |32x-4-0|<系.

So,

role="math" localid="1648029573923" 2<x<2+未2-2<x-2<2+未-20<x-2<未

This means that, when 0<x-2<未, we have

|32x-4-0|=|32x-4|=32(x-2)<18未=18鈭�218=鈭�

So, whenever x鈭�(2,2+未), we also have |32x-4-0|<系.

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Most popular questions from this chapter

Write each of the inequalities in interval notation:

3x+1-2<0.1

For each limit statement in Exercises 41-44, use algebra to find 未or Nin terms of 蔚or M, according to the appropriate formal limit definition.

limx鈫�鈭�x-1x=1, findNin terms of蔚.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of fxas x鈫�cis the value fc.

(c) The limit of fxas x鈫�cmight exist even if the value of fcdoes not.

(d) The two-sided limit of fxas x鈫�cexists if and only if the left and right limits of fxexists as x鈫�c.

(e) If the graph of fhas a vertical asymptote at x=5, then limx鈫�5fx=鈭�.

(f) If limx鈫�5fx=鈭�, then the graph of fhas a vertical asymptote at x=5.

(g) If limx鈫�2fx=鈭�, then the graph of fhas a horizontal asymptote at x=2.

(h) Iflimx鈫�鈭�fx=2, then the graph offhas a horizontal asymptote aty=2.

For each of the following sign charts, sketch the graph of a function f that has the indicated signs, zeros, and discontinuities:

For each of the following sign charts, sketch the graph of a function f that has the indicated signs, zeros, and discontinuities:
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