episilon delta proofs

[Guest]

How do you do an epislon delta proof for
$\displaystyle \lim_{x\to4}\ (3x+4)=16$?

Can someone help me and explain it to me? I don't understand it.

 

[Gene]

I'm not positive what an episilon delta proof is but I think you want
$\displaystyle \lim_{dx\to0}\ (3(4+dx)+4)=16+e$
12+3*dx+4=16+e
3*dx=e
as dx -> 0, e ->0
where dx is delta x and e is episilon.
If I'm wrong, (and no one else pops in) do you have an example from your book? It doesn't make much sense to me when you can just substitue x=4

 

[Guest]

Sorry. I don't have it. It's something that my teacher understands, and a few other mathematicans who are really smart to understand this advanced mathematical proof and ideology. lol. It has something to do with epsilon and delta though. =\.

 

[soroban]

Hello, atse1900!

How do you do an epislon delta proof for: $\displaystyle \lim_{x\to4}\ (3x+4)=16$?

Can someone help me and explain it to me? I don't understand it.

I won't go into the theory behind it, but here's the procedure.

The epsilon statement is: .$\displaystyle |(3x\,+\,4)\,-\,16|\,<\,\epsilon$ . [1]

. . and we must manipulate it into the form: .$\displaystyle |x\,-\,4|\,<\,\delta$ . [2]


We have: .$\displaystyle |3x\,-\,12|\,<\,\epsilon$

Factor: . . $\displaystyle |3(x\,-\,4)|\,<\,\epsilon$

. . . . . . . . . $\displaystyle 3|x\,-\,4|\,<\,\epsilon$

. . . . . . . . . . $\displaystyle |x\,-\,4|\,<\,\frac{\epsilon}{3}$

We have [2] if: $\displaystyle \delta\,=\,\frac{\epsilon}{3}\qquad\leftarrow$(This is the answer)