Prove Limit Using Epsilon-delta Definition...4

[nycmathguy]

Calculus
Section 2.4

I don't need to know this stuff to succeed in Calculus. However, I would like to have your math work reply file away as future study notes and/or reference notes. How about working out 20?

 

[MathLover1]

lim((3/2)x-1/2, x->5)=7

0<|x-5|<δ ; |((3/2)x-1/2)-7|<ε

Step 1:

observe this: ((3/2)x-1/2)-7|=|1/2 (3 x - 1)-7|

Fix any ε>0. To have | (3 x - 1)/2-7| <ε

| (3 x - 1)-14| <2ε
|3x-15|<2ε......both side divide by 3
|x-5|<2ε/3

as ε>0

0<x-5<2ε/3
0<x<2ε/3+5

Step 2:
|x-5|<δ
-δ <x-5<δ
-δ +5<x<δ+5

as δ>0
0<x<δ+5

Observe the relation between |x-c|<δ ; |(f(x))-L|<ε

δ+5 =2ε/3+5
δ =2ε/3

Solution: δ =2ε/3

 

[nycmathguy]

lim((3/2)x-1/2, x->5)=7

0<|x-5|<δ ; |((3/2)x-1/2)-7|<ε

Step 1:

observe this: ((3/2)x-1/2)-7|=|1/2 (3 x - 1)-7|

Fix any ε>0. To have | (3 x - 1)/2-7| <ε

| (3 x - 1)-14| <2ε
|3x-15|<2ε......both side divide by 3
|x-5|<2ε/3

as ε>0

0<x-5<2ε/3
0<x<2ε/3+5

Step 2:
|x-5|<δ
-δ <x-5<δ
-δ +5<x<δ+5

as δ>0
0<x<δ+5

Observe the relation between |x-c|<δ ; |(f(x))-L|<ε

δ+5 =2ε/3+5
δ =2ε/3

Solution: δ =2ε/3

You are amazingly gifted. You learned this stuff 47 years ago and never forgot what to do.

How about the link below? No math threads until Friday.

Prove Limit Using Epsilon-delta Definition...1