A good way to explain $\varepsilon$-$\delta$ for chemistry / biology students? I feel like I have a pretty good way to talk about $\varepsilon$-$\delta$ to physics and engineering students (and possibly students in comp sci). But I am not very sure what I can do for chemistry and biology majors.
For instance, with physics / engineering, I feel like I can talk about designing an apparatus ($f$) whose output is some target value ($L$) with a given tolerance level ($\varepsilon$). How careful do they need to control their input ($\delta$)... etc etc
But when it comes to biology? Or chemistry? I am not sure what a good motivating analogy would be. 
EDIT: Cross-posted on MathEducators.SE.
 A: To explain $f(x) \xrightarrow[x\to a]{} A$ to anyone, I'll choose a graphical approach.  In this way, concepts can be visualized.  As there're more graphs and diagrams in biology and chemistry, they will be understood better than logic symbols.


*

*Plot the curve $y=f(x)$ (If the limit point $a$ doesn't belong to the domain, leave it out.  See the picture below.)

*Draw a horizontal strip centered at $y=A$ with height $2\epsilon$.
$$
\Large\bbox[5px, #FDF, border: 1px solid black]{
y = A \quad \begin{array}{cc} \hline
\updownarrow \epsilon & \phantom{\Rule{6em}{1ex}{0px}} \\ \hdashline
\updownarrow \epsilon & \\ \hline
\end{array}
} \tag{H-strip} \label{fig:hstrip}
$$

*Find a vertical strip of width $2\delta$ so that the deleted curve (point $a$ removed) won't burst out of the rectangle.
$$
\require{extpfeil}
\Newextarrow{\lfa}{5,5}{0x2194}
\Large\bbox[5px, #FDF, border: 1px solid black]{
y = A \quad \begin{array}{c|c|c|c}
& & & \\ \hline
\updownarrow \epsilon & {\small \text{curve}} & {\small \text{inside}} & \\ \hdashline
\updownarrow \epsilon & {\small \text{these}} & {\small\text{grids}} & \\ \hline
& \lfa[\delta]{} & \lfa[\delta]{} &
\end{array}
} \tag{V-strip} \label{fig:vstrip}
$$
If no such $\delta$ can be found, then this $\epsilon$-$\delta$ definition is not satisfied.

*Repeat steps (2) and (3) with smaller $\epsilon$.  If a (thin) \ref{fig:vstrip} satisfying the criterion in step (3) can be drawn no matter how thin the \ref{fig:hstrip} is, then this $\epsilon$-$\delta$ definition is satisfied.

*To consolidate their understanding, ask them to apply these steps to these functions with limit point $a = 0$:


*

*$f(x) = kx$, where $k$ is a constant


$\delta = \frac{\epsilon}{k}$ responds to the $\epsilon$-challenge.


*$f(x) = \frac{x}{x}$ undefined at $x=0$


$\delta = \epsilon$ responds to the $\epsilon$-challenge.


*$f(x) = \frac{\sin x}{x}$ undefined at $x=0$


$\delta = \epsilon$ responds to the $\epsilon$-challenge.


*$f(x) = \sin\frac1x$ undefined at $x=0$


 Infinitely many peaks $(((n+\frac12)\pi)^{-1},1)$ and troughs $(((n-\frac12)\pi)^{-1},-1)$ lie out of the four $\epsilon$-$\delta$ grids.


*$f(x) = x \sin\frac1x$ undefined at $x=0$


 $\delta = \epsilon$ responds to the $\epsilon$-challenge.


Picture from Wiki Commons
