Start the EpsilonDelta  maplet and bring it to the state shown in Figure 1.2.1(a) by the following steps.

In the top row of the interface, enter the function as 3*x - 4, and enter $a\=3$, and $L\=5$ in the appropriate windows.

Set the plot ranges to $\mathrm{xmin}\=0$, $\mathrm{xmax}\=5$, $\mathrm{ymin}\=0$, and $\mathrm{ymax}\=10$.

Set $\mathrm{\ϵ}\=0.50$ and $\mathrm{\δ}\=0.50$ (use the slider to come close to these values).

Click on the Plot button at the bottom of the Maplet window.

Change the value of $\mathrm{\δ}$ to 0.05.

Now the blue horizontal strip falls entirely within the red lines. This value of $\mathrm{\δ}$ does satisfy the conditions in the definition of the limit.

But, this is not enough! The definition states that it must be possible to find such a δ for every $\mathrm{\ϵ}\>0$. At present, this property has been shown true only for $\mathrm{\ϵ}\=0.5$.

Find (exactly or approximately) the largest value of $\mathrm{\δ}$ for which the conditions of the definition of the limit are satisfied for $\mathrm{\ϵ}$  = 0.50.

Repeat step (3) for $\mathrm{\ϵ}$  = 0.25. (Try $\mathrm{\δ}$  = 0.1 and $\mathrm{\δ}$ = 0.05 to get started.)

Is it possible to conjecture a general rule for selecting $\mathrm{\δ}$ for any given $\mathrm{\ϵ}$?