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Chapter 1: Q. 30 (page 88)

Consider the limit values below:

limx鈫�2-f(x)=3,limx鈫�2+f(x)=3andf(2)=0

The strategy is to sketch the graph of the function having the above limit values

Short Answer

Expert verified

The given limit values are

limx鈫�2-f(x)=3,limx鈫�2+f(x)=3andf(2)=0

Step by step solution

01

Step 1. Given 

The given limit values are

limx鈫�2-f(x)=3,limx鈫�2+f(x)=3andf(2)=0

02

Step 2. Breakup point

The left-hand limit,limx->2-f(x)=3represents that the function value approaches to 3 as x approaches to 2 from the left side and the right-hand limit, limx->2+f(x)=3represents that the function value approaches to 3 as x approaches to 2 from the right side.

Thus, the limit of the function at x = 2is limx->2f(x)=3

But the value of the function at x = 2 is given 0. That is, f(2) = 0 .

This means that the function value is not same as limit value. This implies that the function is not continuous at x = 2 becauselimx->2-f(x)鈭�f(2)

Hence, the function is discontinuous at x = 2 . So, there is a break in the graph at point

x = 2

Comment

03

Step 3. Graph obtained 

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Most popular questions from this chapter

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.

limx鈫�43e1.7x+1

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:

limh鈫�0(3+h)2-32h.

limx鈫�11-cos(x-1)x
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