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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

How would you "remove the discontinuity" of f? In other words, how would you define f(2) in order to make f continuous at 2?

f ( x ) = x 2 x 2 x 2

To determine

To define: The function f(x)=x2x2x2 at the point x=2 in order to make f continuous at 2.

Explanation

Definition used: “A function f is continuous at a number a if limxaf(x)=f(a)”.

Note 1: “The limit may be infinite or some finite value when both the numerator and the denominator approach to 0.”

Calculation:

Define f(2) in order to make f continuous at 2.

Substitute 2 for x in f(x)=x2x2x2 as follows,

f(2)=(2)2(2)2(2)2=4422=00

Thus, f(x)=x2x2x2 is not defined at x=2. So, remove the discontinuity at x=2 by redefining a function f at an isolated number 2.

By note 1, compute the limit of the function f(x) as x approaches to 2 but x2.

limx2f(x)=x2x2x2=x22x+x2x2=x(x2)+(x2)x2=

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