   Chapter 10.3, Problem 2E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In Exercises 1–4, complete the given sentence.The closed-form function f ( x ) = 1 x 2 − 1 iscontinuous for all x expect _ _ _ . [HINT: See Quick Example 3.]

To determine

To fill: The blank provided in the statement “The closed-form function f(x)=1x21 is continuous for all x except _______.”

Explanation

Consider the function f(x)=1x21.

The function f(x) is continuous when:

a) The limxaf(x) exists

b) limxaf(x)=f(a)

Here the function cannot be defined when the denominator is equal to zero.

It means x21=0.

So, try to find the value of x where the function is undefined:

x21=0x=1=±1

Check for continuity at x=1:

Consider the right limit:

limx1+f(x)=limx1+(1x21)

Apply continuity of closed form function theorem:

limx1+(1x21)=1(1+)21=10+=+

Consider the left limit:

limx1f(x)=limx1(1x21)

Apply continuity of closed form function theorem:

limx1(1x21)=1(1)21=10+=

The right side limit and left side limit is not definite

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