Help needed with a specific epsilon-delta limit proof

[sergey90]

Homework Statement

limit[1/(x-2)^3]=-inf as x->2

Homework Equations

The Attempt at a Solution

2-delta<x<2 1/(x-2)^3 < M
-delta<x-2<0 (x-2)^3>1/M
(-delta)^3<(x-2)^3<0

=>(-delta)^3=1/M=>-delta=croot(1/M)=>delta=-croot(1/M) ...huh? how could delta be negative?

 

[micromass]

I have no idea what you just did.

What are these three equations?? What is M??

 

[sergey90]

its the epsilon delta proof for infinite limits. Its just how they are proved generally. M is any negative number

 

[micromass]

I still have no idea what you did, sorry. I can see delta's and M's and stuff, but I have no idea what you're doing.

Can you write in words what you're doing between every step? Write what you want to prove. Just write some text to guide the reader.

Also, if M is negative, then -\sqrt[3]{1/M} is positive...

 

[STEMucator]

Homework Statement

limit[1/(x-2)^3]=-inf as x->2

Homework Equations

The Attempt at a Solution

2-delta<x<2 1/(x-2)^3 < M
-delta<x-2<0 (x-2)^3>1/M
(-delta)^3<(x-2)^3<0

=>(-delta)^3=1/M=>-delta=croot(1/M)=>delta=-croot(1/M) ...huh? how could delta be negative?

You want to prove :

lim_{x→2} \frac{1}{(x-2)^3} = -∞

So you want to use this statement :

\forallM>0, \existsδ>0 | 0 < |x-2| < δ \Rightarrow f(x) < M

Start with f(x) < M and massage it to find a suitable δ.