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)