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Chapter 1: Q. 7 (page 152)

Fill in the blanks to complete each of the following theorem statements:

A function f can change sign from positive to negative, or vice versa, atx=c only if f(x)is , , or atx=c.

Short Answer

Expert verified

A function f can change sign from positive to negative, or vice versa, atx=conly if f(x)is zero, undefined, or discontinuous atrole="math" localid="1648288689517" x=c.

Step by step solution

01

Step 1. Given information. 

The given incomplete statement is folowing.

A function f can change sign from positive to negative, or vice versa, atx=conly if f(x)is , , or atx=c.

02

Step 2. Explanation. 

A function f can change sign from positive to negative, or vice versa, atx=conly if f(x)is zero, undefined, or discontinuous atrole="math" localid="1648288720072" x=c.

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Most popular questions from this chapter

limx鈫�蟺1cosec(蟺-x)

Sketch the graph of a function f described in Exercises 27鈥�32, if possible. If it is not possible, explain why not.

f is left continuous at x = 2 but not continuous at x = 2, and f(2) = 3.

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.

Calculate each of the limits:

limh鈫�0(1+h)3-13h.

For each functionf graphed in Exercises23鈥�26, describe the intervals on whichf is continuous. For each discontinuity off, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.
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