# The Definition of the Limit

Recall that the limit is defined like this.

*We say that the limit of a function \(f\) at \(x_0\) is \(L\)
if, given any positive number \(\epsilon\), we can choose a positive number
\(\delta\) small enough that
\[\vert x-x_0\vert < \delta\ \textit{implies}\ \vert f(x)-L\vert < \epsilon.\]
*

Below you can see a diagram where we have marked a value for \(x_0\) using a dotted vertical line. It seems likely that our limit for \(f\) at \(x_0\) is \(L=f(x_0)\). There are two horizontal lines marking \(L+\epsilon\) and \(L-\epsilon\). The red box shows the area containing all values of \(f(x)\) with \(\vert x-x_0\vert < \delta \). If we can choose \(\delta\) small enough to make the box lie between the horizontal lines, then that is a \(\delta\) that satisfies the definition.

## Questions

- Given \(x_0=1.8\) and \(\epsilon=0.1\), give one value of delta that forces \(\vert f(x)-L\vert < \epsilon\).
- Again with \(x_0=1.8\) and \(\epsilon=0.1\), give a second value of delta that forces \(\vert f(x)-L\vert < \epsilon\).
- Changing \(x_0\) to \(2.2\) and leaving \(\epsilon=0.1\), give a value of delta that forces \(\vert f(x)-L\vert < \epsilon\).
- Changing \(x_0\) to \(1.0\) and leaving \(\epsilon=0.1\), give a value of delta that forces \(\vert f(x)-L\vert < \epsilon\).