Question

each of exercises 15–30 gives a function ƒ(x) and numbers l, c, and e 7 0. in each case, find an open interval about c on which the inequality 0 ƒ(x) - l 0 6 e holds. then give a value for d 7 0 such that for all x satisfying 0 6 0 x - c 0 6 d the inequality 0 ƒ(x) - l 0 6 e hol

Answer 1

The given inequality holds for the open interval (2.97,3.03)

It is given that

f(x)=6x+7

cL=25

c=3

ε=0.18

We have,

|f(x)−L| = |6x+7−25|

          = |6x−18|

          = |6(x−3)|

          = 6|x−3|

Now,

6|x−3| <0.18  then |x−3|<0.03 ----->−0.03<x-3<0.03---->2.97<x<3.03

the given inequality holds for the open interval (2.97,3.03)

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Although part of your question is missing, you might be referring to this full question: For the given function f(x) and values of L,c, and ϵ0, find the largest open interval about c on which the inequality |f(x)−L|<ϵ holds. Then determine the largest value for δ>0 such that 0<|x−c|<δ→|f(x)−|<ϵ.

f(x)=6x+7,L=25,c=3,ϵ=0.18

 

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