General formula to find δ, given ε, c, f(x), and L

[TylerH]

I noticed there's a pattern to the special cases f(x)=ax+b and f(x)=x^2:
$$lim_{x\rightarrow c}ax+b=L, \delta=\frac{\epsilon}{a}$$
$$lim_{x\rightarrow c}x^2=L, \delta=\frac{\epsilon}{2c+1}$$
I noticed that a is the derivative of ax+b, and 2x(2c) is the derivative of x^2, but how do I justify the + 1 part of the special case for x^2?

Is there any formula for a general case?

 

[TylerH]

Is there a pattern, or am I seeing one that doesn't exist?

 

[LCKurtz]

Of course, for such problems, there isn't a unique δ that works for a given ε, since if you have one value for δ, any smaller one will work. So the problem is well defined if you are looking for the largest δ that will work for a given ε, which is what you have done in the linear case.

The short answer to your question is no. You can, in principle, always find the minimum δ, but it won't always be easy and certainly you won't find a nice pattern, even for polynomials.

 

[TylerH]

You can, in principle, always find the minimum δ, but it won't always be easy and certainly you won't find a nice pattern, even for polynomials.

Do you mean "[...] find the largest δ, [...]"?

Is there a general way to find the largest delta given f(x) and the largest epsilon?

 

[LCKurtz]

Of course, for such problems, there isn't a unique δ that works for a given ε, since if you have one value for δ, any smaller one will work. So the problem is well defined if you are looking for the largest δ that will work for a given ε, which is what you have done in the linear case.

The short answer to your question is no. You can, in principle, always find the minimum δ, but it won't always be easy and certainly you won't find a nice pattern, even for polynomials.

Do you mean "[...] find the largest δ, [...]"?

Is there a general way to find the largest delta given f(x) and the largest epsilon?

Yes, that is clearly a typo isn't it, given the emphasis on largest above. I meant maximum when I wrote minimum. So, yes, you can do it in principle but you may have to resort to numerical methods but no, it won't give you any nice pattern. And you don't mean "given f(x) and the largest epsilon". Epsilon is specified first, then you can find delta depending on it.