Limits with [math]\epsilon[/math] and [math]\delta[/math].[br][br]Use of mapping diagrams to visualize the definition of limits is not uncommon for real or complex analysis.[br]The strength of these visualizations is the power it gives to understanding how, where, and why the [math]\epsilon's[/math] and [math]\delta's[/math] are chosen.[br][br]First [math]\epsilon>0[/math] is chosen - arbitrarily - to identify an open region in the target containing the proposed limit number, [math]L[/math].[br]Then a [math]\delta>0[/math] is chosen to identify an appropriate open region in the domain, [math] \{ z:|z-z_0| \lt \delta \}[/math] containing the given point of interest, [math]z_0[/math]. [br]Finally it must be demonstrated that for any [math]z[/math] with [math]|z-z_0|<\delta[/math] in the domain , the value [math]f(z)[/math] satisfies[math]|f(z)-L|<\epsilon[/math].[br][br][br]The following GeoGebra examples allow you to investigate the limit definition both [b]when it fails and when it succeeds.[/b][br]