Proving the Limit of a Function Using Epsilon-Delta Definition

[azizlwl]

Homework Statement

If f(x)=x² prove that \lim_{x \to 2} f(x)= 4
Solution given:
We must show that given any ε >0, find δ >0 such that |x²-4|<ε when 0<|x-2|<δ

Choose δ≤1 so that <|x-2|<1
-----------------------------------------------
Confuse between the word 'find' and 'choose'.

 

[LCKurtz]

Homework Statement

If f(x)=x² prove that \lim_{x \to 2} f(x)= 4
Solution given:
We must show that given any ε >0, find δ >0 such that |x²-4|<ε when 0<|x-2|<δ

Choose δ≤1 so that <|x-2|<1
-----------------------------------------------
Confuse between the word 'find' and 'choose'.

$|x^2-4|=|(x+2)(x-2)|=|x+2|\cdot |x-2|$. So if $|x-2|<1$ how big can $|x+2|$ be? Then once you figure that out, how much smaller than 1 does $|x-2|$ need to be to make the whole thing less that $\epsilon$?

 

[HallsofIvy]

You can "find" many values of \delta that will work and then "choose" one of those. That is the same as "finding" a value.

 

[azizlwl]

Thanks. My confusion must be interpreting the word "find" as calculate in usual mathematics or physics problems.