- #1

Luscinia

- 17

- 0

## Homework Statement

Calculate the value of the limit and justify your answer with the ε-δ definition of the limit.

lim (x->1) x

^{2}

## Homework Equations

My professor gave us the hint that we have to take δ as 0<δ≤ k

_{0}so that δ(ε)=min{k

_{0},ε/ (k

_{0}+2)}

I'm guessing that k

_{0}is meant to be any number though it's usually 1?

## The Attempt at a Solution

I'm trying to relate |f(x)-1|<ε to |x-a|<δ

|x

^{2}-1| < ε

-ε < x

^{2}-1 <ε

-ε+1 < x

^{2}<ε+1

(-ε+1)

^{1/2}-1 < x-1 < (ε+1)

^{1/2}-1

I'm not sure what to do from here. I'm guessing that

δ=(-ε+1)

^{1/2}-1 or (ε+1)

^{1/2}-1

but I have no clue where to go from there.

I've tried doing this another way as well to make use of a k

_{0}from my professor's hint

|x

^{2}-1| < ε

-ε < x

^{2}-1 <ε

-ε+1 < x

^{2}<ε+1

-ε+1 < x x < ε+1 where if we assume that |x|<k

_{0}=1, we can then assume that -ε+1 < k

_{0}x < ε+1

(-ε+1)/k

_{0}- 1 < x-1< (ε+1)/k

_{0}- 1

This still doesn't give me what my professor hinted at though. (I don't know what to do after that.)

Also, what does the min{_____,_______} mean?