1. Limit of a constant

if $\displaystyle f(x)=c$ where c is a constant and it is asked to find

$\displaystyle \lim_{x \to a} f(x) = L$

According to the definition we can take

$\displaystyle \forall \epsilon >0 , \exists\delta>0$

$\displaystyle 0<|x-a|<\delta \implies |f(x)-L| < \epsilon$

my question is whether $\displaystyle \delta$ and $\displaystyle \epsilon$ are independent from each other

if $\displaystyle f(x)=c$ where c is a constant and it is asked to find

$\displaystyle \lim_{x \to a} f(x) = L$

According to the definition we can take

$\displaystyle \forall \epsilon >0 , \exists\delta>0$

$\displaystyle 0<|x-a|<\delta \implies |f(x)-L| < \epsilon$

my question is whether $\displaystyle \delta$ and $\displaystyle \epsilon$ are independent from each other

Of course not: $\displaystyle \delta$ almost always depends of $\displaystyle \epsilon$ (and on the function and on what point is x tending

to, of course), but in this case it's trivially true that $\displaystyle L=c\,,\,\,\delta=$ whatever , as you can easily check.

Tonio

3. The largest possible value of $\displaystyle \delta$ depends on $\displaystyle \epsilon$ but $\displaystyle \delta$ can be any number smaller than that largest possible value. For this particular problem, the largest possible value of $\displaystyle \delta$ is infinity (there is no largest possible value).