I am teaching elementary analysis and introducing the concept of $\epsilon$-$\delta$ definition of the limit to first time learners. For example, we take $\displaystyle\lim_{x \to 2} (2x - 1) = 3$, then we say by definition:
For every $\epsilon >0$, there exists a $\delta > 0$ such that if $0<|x-2|<\delta$, then $|2x-1-3|<\epsilon$.
We can go on and say that $|2x-4| = 2|x-2| < \epsilon$. From which we can see that: $|x-2| < \frac{\epsilon}{2}$. Hence, we can choose $\delta = \frac{\epsilon}{2}$.
Now all this seems very very abstract to a first time learner, and so I always supplement with a table of values (which is equivalent to the $\epsilon$-$\delta$, but not too obvious to the first learner.)
For example:
\begin{array}{ccc} x &1.8 & 1.9 & 2 & 2.1 & 2.2 & 2.3 \\ y & 2.6 & 2.8 & \# & 3.2 & 3.4 & 3.6 \end{array}
I intentionally blocked off $y=3$ to argue that the function need not be defined at that point for the limit to exist. Of course, we can always just put 3 for simplicity. Finally, we see that for every $0.1$ increment in $x$, there is a $0.2$ increment in $y$. OR, you can say for every $0.2$ increment in $x$, there is a corresponding change of $0.4$ in $y$. (and other analogies.) Hence, learners can easily see the $\epsilon/2$ in the ratios of these increments, and everyone is happy.
Now, my question is, I want to extend this heuristic when I teach the $\epsilon$-$\delta$ definition now, for functions that are non-linear. Say, $\lim_{x \to 2} x^2 - 1 = 3$. I am lost. (and perhaps, became a victim of myself trying to make this simpler.)
Any insights?