Proof ε-δ: Epsilon Delta Proof of lim x->a of ((3x²-3a²)/(x-a)) = 6a

[Dani4941]

I’m going to say from the beginning that I need to hand this problem in. I'm not looking for the answer, I think I already have it, just want a critical eye.

I need someone to look over this problem and tell me if it's good. Not just if it's right but if it's perfect. I always get the problem right then get minus points because I didn't explain it enough.

So here it is

Give an ε-δ proof of
lim x->a of ((3x²-3a²)/(x-a)) = 6a

Proof: Note |f(x) – a| = |f(x) – 6a| = | (3x²-3a²-6ax-6a²)/(x-a) | = | (3(x²-2ax+a²)/(x-a)) | = |3x-3a|

|3x-3a|<ε (get rid of 3 to make it smaller) |x-a|<ε when |x-a|<δ let δ=ε


Given ε>δ let δ=ε

0<|x-a|<δ then
|f(x) – a| = |3x-3a|<δ=ε

Therefore
lim x->a of ((3x²-3a²)/(x-a)) = 6a

 

[scottie_000]

What you really need to say is let d = e/3
Then you get |x-a|<d ==> |3x-3a|< 3*d =e, which is what you needed.
Up to there it was fine though

 

[Dani4941]

ooo Thanks. That was a pretty large mistake.