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Chapter 1: Q. 12 (page 119)

Each function in Exercises 9鈥�12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

limx鈫�2-f(x)=-鈭�,limx鈫�2+f(x)=鈭�,f(2)=3.

Short Answer

Expert verified

The type of discontinuity is an infinite discontinuity and there is not any one-sided continuity.

The graph of f is

Step by step solution

01

Step 1. Given Information. 

The given function islimx鈫�2-f(x)=-鈭�,limx鈫�2+f(x)=鈭�,f(2)=3.

02

Step 2. Describing the discontinuity. 

From the function, we can depict that f(x)has infinite discontinuity because both of limx鈫�2-f(x)andlimx鈫�2+f(x)areinfinite.

03

Step 3. Describing one-sided continuity at x=c.

There is not any one-sided continuity at x=2because limx鈫�2-鈮�f(2)andlimx鈫�2+鈮�f(2).

04

Step 4. Graph of f.

The graph of fis

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Most popular questions from this chapter

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.

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:

limx鈫�01sec-1x.

In Exercises 39鈥�44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

f(x)=x2,ifx<24,ifx=22x+1,ifx>2.

Each function in Exercises 9鈥�12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

limx鈫�2-f(x)=2,limx鈫�2+f(x)=1,f(2)=1.
See all solutions

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