   Chapter 2.5, Problem 44E

Chapter
Section
Textbook Problem

In Exercises 39-44, determine the values of x, if any, at which each function is discontinuous. At each number where f is discontinuous, state the condition(s) for continuity that are violated.44. f ( x ) = { x 2 − 1 x + 1 if  x ≠ − 1 1 if  x = − 1

To determine

The value of x at which the function f(x)={x21x+1ifx11ifx=1 is discontinuous.

Explanation

Definition used:

The function will be continuous when the value of left hand limit and the value of right hand limit is equal to the value of the function at that point and value of that function should satisfy that point.

A function f is continuous at x=a when it follows the following three conditions,

(1) f(a) is defined.

(2) limxaf(x) exists.

(3) limxaf(x)=f(a).

Calculation:

The function is f(x)={x21x+1ifx11ifx=1 (1)

The graph of the function is shown below in Figure 1,

Figure (1)

Substitute 1 for x in equation (1) to find the left hand limit,

limx1f(x)=limx1(x21x+1)=limx1((x+1)(x1)x+1)((a2b2)=(a+b)(ab))=limx1(x1)(Canceloutthecommonfactor)=11

Further solve the above equation,

limx1f(x)=2

Substitute 1 for x

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