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Chapter 1: Q. 57 (page 98)

For each limit in Exercises 55鈥�64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.

limx鈫�鈭�3xx+1=3,蔚=0.5,findsmallestN>0

Short Answer

Expert verified

The required value ofN=5

Step by step solution

01

Step 1. Given Information   

The given function isf(x)=3xx+1

02

Step 2. Explanation  

From the given function, we have, c=鈭�,L=3

The limit expression can be written as a formal statement as below,

For all epsilon positive, there is some N positive such that if x鈭�(N,鈭�)

Then localid="1648048169455" role="math" 3xx+1鈭�(3-蔚,3+蔚)

Now the smallest value of N is given by,

localid="1648047368652" 3xx+1=3-0.53xx+1=2.53x=2.5x+2.50.5x=2.5x=5

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Most popular questions from this chapter

Write each of the inequalities in interval notation:

x2-1+3<0.5

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.

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�3(x+5)=8

Calculate each of the limits:

limh鈫�0(-1+h)3-(-1)3h.

State what it means for a function f to be right continuous at a point x = c, in terms of the delta鈥揺psilon definition of limit.
See all solutions

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