Epsilon delta to prove continuity

[james.farrow]

I have an example bit I can't quite follow it...?

Use epsilon -delta definition of continuity to prove f(x) = 3x^2 - x is continuous at x=2

Ep > 0 and delta > 0 in terms of Ep

f(x) -f(2) = 3x^2 - x -(3*2^2 -2)

f(x) - f(2) = 3x^2 -x - 10
f(x) - f(2) = (3x + 5)(x - 2)

So far so good - but now can someone explain what happens please...!

James

 

[arildno]

Well, we may, for example, rewrite 3x+5 as 3(x-2)+11.

Now, set d=x-2

We then have:
|f(x)-f(2)|=|(3d+11)d|<=3|d|^2+11|d|<=14|d|, if |d| is tiny enough, specifically, when |d|<1 (i.e, when x is between 1 and 3)

Now, can you make |d| so small that given any e, 14|d| will be smaller than e?

 

[tiny-tim]

Hi James!

(have a delta: δ and an epsilon: ε and try using the X² tag just above the Reply box )

Use epsilon -delta definition of continuity to prove f(x) = 3x^2 - x is continuous at x=2

Ep > 0 and delta > 0 in terms of Ep

f(x) -f(2) = 3x^2 - x -(3*2^2 -2)

You're probably wondering "wherever does δ come into it?? "

You needed to start with "f(2 + δ) -f(2) = …" ​

 

[james.farrow]

Thanks for the replies lads - I appreciate it! What I need is an explanantion of the whole epsilon delta thing really from start to finish.

I don't understand it to be honest and I need to so I can apply it to other functions etc

Many thanks

James

 

[jgens]

If you do a search, epsilon-delta proofs have been explained many times. See this thread for example: https://www.physicsforums.com/showthread.php?t=54175&highlight=limit. This thread also has some explanations: https://www.physicsforums.com/showthread.php?t=315877

Hope this helps!

 

[arildno]

Thanks for the replies lads - I appreciate it! What I need is an explanantion of the whole epsilon delta thing really from start to finish.

I don't understand it to be honest and I need to so I can apply it to other functions etc

Many thanks

James

Start with the first thing you don't understand.