Prove Limit Using Epsilon-delta Definition...3

[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 28?

 

[MathLover1]

28.

lim((6+x)^(1/8),x->-6^+)=0

want to find delta so that when |x-(-6)|< delta, |(6+x)^(1/8)-(-6)|<ϵ

|x+6|< delta, |(6+x)^(1/8)+6|<ϵ

start here
|(6+x)^(1/8)+6|<ϵ
ϵ>|(x + 6)^(1/8) + 6|

proof:

suppose ϵ>0 is given
choose δ = ϵ=|(x + 6)^(1/8) + 6|
δ = |(x + 6)^(1/8) + 6|

assume δ <ϵ

|x+6|<|(x + 6)^(1/8) + 6|

-6<=x<1.28168

so,

x = -6, δ>0, ϵ>|(x + 6)^(1/8) + 6|

 

[nycmathguy]

28.

lim((6+x)^(1/8),x->-6^+)=0

want to find delta so that when |x-(-6)|< delta, |(6+x)^(1/8)-(-6)|<ϵ

|x+6|< delta, |(6+x)^(1/8)+6|<ϵ

start here
|(6+x)^(1/8)+6|<ϵ
ϵ>|(x + 6)^(1/8) + 6|

proof:

suppose ϵ>0 is given
choose δ = ϵ=|(x + 6)^(1/8) + 6|
δ = |(x + 6)^(1/8) + 6|

assume δ <ϵ

|x+6|<|(x + 6)^(1/8) + 6|

-6<=x<1.28168

so,

x = -6, δ>0, ϵ>|(x + 6)^(1/8) + 6|

Lord Jesus, this Epsilon-delta stuff is not sinking into my brain. Are you willing to do a few more for me to study when time allows? Thank God I don't have to do know this stuff to learn calculus as a whole. I need you to break down each step with words from start to finish. Also, no further questions will be posted today because there are 3 unanswered threads in the Calculus forum.

 

[nycmathguy]

28.

lim((6+x)^(1/8),x->-6^+)=0

want to find delta so that when |x-(-6)|< delta, |(6+x)^(1/8)-(-6)|<ϵ

|x+6|< delta, |(6+x)^(1/8)+6|<ϵ

start here
|(6+x)^(1/8)+6|<ϵ
ϵ>|(x + 6)^(1/8) + 6|

proof:

suppose ϵ>0 is given
choose δ = ϵ=|(x + 6)^(1/8) + 6|
δ = |(x + 6)^(1/8) + 6|

assume δ <ϵ

|x+6|<|(x + 6)^(1/8) + 6|

-6<=x<1.28168

so,

x = -6, δ>0, ϵ>|(x + 6)^(1/8) + 6|

Finally got a reply from Jenn's staff.

 

[MathLover1]

it was about time

 

[nycmathguy]

it was about time

Check your PM.