# How to choose a limit for the epsilon delta proof?

The epsilon delta proof allows us to prove that a 'given' limit is actually the limit. In the sense that, we can show using that if a limit for a sequence/ function exist, then that limit is a unique one. But, how does one find a limit following the epsilon delta definition or do we need something more?

• FYI: Eventually, you'll learn about Cauchy sequences, where what matters is that terms get arbitrarily-close to each other (roughly speaking ) rather than arbitrarily-close to a predetermined value $L$. In appropriate contexts, this is enough to ensure convergence to a limit without having to know the limiting value ahead of time.
– Blue
Jul 24, 2022 at 8:48
• Mmm I have. But that is only in complete metric space. Good point. Jul 24, 2022 at 8:50
• "complete metric space" = "appropriate context". :)
– Blue
Jul 24, 2022 at 8:52
• Your question seems to boil down to "How do you compute limits?" (as "finding the $L$" in an $\varepsilon$-$\delta$ proof is precisely the task of computing the limit). This is an incredibly broad question. I don't see how it can be reasonably answered---even the accepted answer her only gives very vague advice. Jul 24, 2022 at 15:38
• There are many techniques for computing limits, and many theorems about limits. Generally, we use those theorems and techniques, and are taught to use them in analysis classes. But there is not (nor can there be) a technique which will allow you to compute every possible limit. Jul 24, 2022 at 19:45

All of this can be done on scrap paper and, so to speak, is hidden away from public view. We don't publicly tell everyone that we found a limit by L'Hopital's rule. We just tell them what the limit is and prove that it is the limit via the golden standards of proof-writing, whether it be by an $$\epsilon-\delta$$ argument or by making use of other theorems (by the way, making use of theorems to prove a limit where it is justified is perfectly rigorous.)