Can $\varepsilon$-$\delta$ definitions be used to find a limit or only to verify? so  I was wondering if there is any part of the $\varepsilon$-$\delta$ definition of the limit that offers any insight on how to find the limit of a function, or if this is something you are supposed to guess at based on the function itself or other elementary functions whose limits you have found. 
For example, if I have the simple function
$$\lim_{x\rightarrow 0} \rvert x \lvert$$
Can I use the epsilon delta definition to figure out the limit? Or do I just have to make a reasonable guess that it is 0 and then use the epsilon delta definition to prove it?
Thanks
 A: In computing limits, writing something like $\lim\limits_{x\to3}\dfrac{x-3}{x^2-9}$, one uses algebra to reduce the problem to $\lim\limits_{x\to3}\dfrac{1}{x+3}$, one uses continuity of the reciprocal function and the polynomial function in the denominator to justify plugging in $3$ at that point.  The point at which $\varepsilon$-$\delta$ proofs enter the process would be in proving that things like polynomial functions and the reciprocal function are continuous.
A: The $\epsilon$-$\delta$ definition will not tell you how to find a limit, because it depends on already knowing the limit. That is, say we have a function $f(x)$. Then, $\lim\limits_{x\to p} f(x) = P~$ if...
$$\forall\epsilon > 0. ~\exists \delta > 0 \ni \forall x \in\text{Dom}(f).~ |x - p| < \delta \implies |f(x) - P | < \epsilon$$
To put it in more easily understood terms: The $\epsilon$-$\delta$ definition of a limit says that $P$ is a limit of $f(x)$ if and only if $f(x)$ gets closer to $P$ as $x$ gets closer to $p$ (the $\epsilon$ and $\delta$ here are like measurements of distance). This definition only gives us a criterion for figuring out whether or not $P$ is a limit—but we have to first know $P$ in order to use the definition.
Theoretically, you could use the $\epsilon$-$\delta$ definition to infer a limit, but that'd be the equivalent of just looking at $f(x)$ itself (with some extra, explicit parameters that may make things more difficult).
A: According my knowledge, language is only used to prove that the limit is, for example, A, however, it is not used to compute what is the limit.
