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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Which of the following functions .f has a removable discontinuity at a? If the discontinuity is removable, find a function g that agrees with f for xa and is continuous at a.

(a) f ( x ) = x 4 1 x 1 , a = 1

(b) f ( x ) = x 3 x 2 2 x x 2 , a = 2

(c) f ( x ) = sin x , a = π

(a)

To determine

To check: Whether the function f(x)=x41x1 has a removable discontinuity at a=1 or not and also check if it is removable discontinuous then find the function g(x) with agrees with f(x) for x1 and which is continuous at x=1.

Explanation

Removable discontinuity:

The function has a removable discontinuity at x=a means that, “The limit exists at x=a and the actual value of f(x) at x=a may or may not be defined. If f(a) is defined, then limxaf(x)f(a)”.

Calculation:

Consider the function is, f(x)=x41x1.

Expand the function by using elementary algebra,

f(x)=(x2)2(1)2x1=(x2+1)(x21)(x1)=(x2+1)(x1)(x+1)(x1)=(x2+1)(x+1)

Then, f(x)=x3+x2+x+1.

To check the existence of the limit as x approaches 1.

limx1f(x)=limx1(x2+1)(x+1)=(12+1)(1+1)=(2)(2)=4

Hence the limit exists at x=1

(b)

To determine

To check: Whether the function f(x)=x3x22xx2 has a removable discontinuity at a=2 or not also check if it is removable discontinuous then find the function g(x) with agrees with f(x) for x2 and which is continuous at x=2.

(c)

To determine

To check: Whether the function f(x)=sinx is removable discontinuous at a=π or not and also check if it is removable discontinuousthen find the function g(x) with agrees with f(x) for xπ and which is continuous at x=π.

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