Delta-Epsilon Definition of a Limit...3

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Calculus
Section 1.7

Can you work out 31 in step by step fashion as a guide for me to do a few more on my own?

Screenshot_20220422-230705_Samsung Notes.jpg
 
31.

Prove
upload_2022-4-23_16-25-7.jpeg


We need to manipulate the
upload_2022-4-23_16-25-51.jpeg
to show that |x+2|<something to set delta equal to that term:

upload_2022-4-23_16-26-57.jpeg


Since we cannot have a x term with epsilon, we let δ=1 and solve for the value x+2 would be:
0<|x+2|<1

−1<x+2<1

−1−4<x+2−4<1−4

−5<x−2<−3

Here, we choose the larger value since if we chose the smaller value, the -5 would not be included, so:
|x−2|<5

Therefore,
|x+2|<ε5

Proof:

upload_2022-4-23_16-29-20.jpeg
 
31.

Prove View attachment 2702

We need to manipulate the View attachment 2703 to show that |x+2|<something to set delta equal to that term:

View attachment 2704

Since we cannot have a x term with epsilon, we let δ=1 and solve for the value x+2 would be:
0<|x+2|<1

−1<x+2<1

−1−4<x+2−4<1−4

−5<x−2<−3

Here, we choose the larger value since if we chose the smaller value, the -5 would not be included, so:
|x−2|<5

Therefore,
|x+2|<ε5

Proof:

View attachment 2705

OMG! This is like Chinese backwards. Sorry but more time is needed to review this craziness.
 

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