Note : my question does not deal with the limit proof proper, but with the preliminary work the goal of which is to find a reasonable candidate for "delta".
Suppose I want to prove that $L$ is the limit of a function $f$ as $x$ approaches a.
What I have to prove is that, under the assumption that $\epsilon\gt0$, for some value of $\delta\left(\gt0\right)$, the conditional statement :
|x-a| $\lt\delta$ --> |L-f(x)|$\lt\epsilon$
is true.
Is it reasonable , in general, to hope finding some fitting delta value by assuming the negation of the consequent?
Are there special cases in which this strategy works?
Does it count amongst standard strategies to find an appropriate delta value?
My teacher in first year of college taught us his "secret trick, we weren't supposed to share with anybody, ever" (hope he doesn't find out): Work backwards. Get a $\delta$, and work out a suitable $\epsilon$ on scratch paper. An a clean sheet, turn the work around: Start with $\epsilon$, and derive some horrible expression (possibly using some mystifing steps in the middle) for a suitable $\delta$. Very often works to give a clean proof. Critical is to be able to undo the steps!