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Chapter 1: Q. 59 (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�lnx=鈭�,M=100,findsmallestN>0

Short Answer

Expert verified

The required value ofN=e100

Step by step solution

01

Step 1. Given Information 

The given function isf(x)=lnx

02

Step 2. Explanation   

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

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

For all M positive, there is some N positive such that ifx鈭�(N,鈭�)

Thenlnx鈭�(M,鈭�)

Now the smallest value of N is given by,

lnx=100x=e100

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Most popular questions from this chapter

Write each of the inequalities in interval notation:

0<|x+3|<0.05

Use algebra to find the largest possible value of 未 or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.
Ifx>N, then1鈭�2x<鈭�500

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.

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.

Calculate each of the limits:

limx鈫�01sec-1x.
See all solutions

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