We have a discussion about left side and right side behavior, and whether or not the table comes to a conclusion. For this example we would say yes, there is a conclusion. The limit is 0.6. You might start getting questions about Mean Girls, the greatest or worst thing to happen to the study of limits. Some kid might ask if the table was necessary, couldn't we just substitute 4 directly?
Save these questions and move on to the next example. Or, show this YouTube clip:
Coincidentally, the garbage function in the background has an asymptote at zero, making the famous line technically accurate.
On to example two, something like the lim as x > -5 of f(x) = 7 / (x + 5):
Now the left side and right side don't agree with one another, causing the limit not to exist. Pause 30 seconds for the "OHHHHHHHHHHHHHHHH" to make its way around the room.
The biggest reason I like the desmos approach is that you very quickly can see the graph that is related to the table. Simultaneously you can discuss how a table would describe the failure of a limit to exist and the graph agrees. This function clearly misbehaves at x = -5, now we have a sophisticated way to talk about it. The exact reason I covered Rational Functions the way I did in the fall was for the discussion we had about limits.
What desmos brings to the party here is the ability to talk about limits as a graphically and table-based concept simultaneously. Some students may not be able to visualize "left" vs "right" based simply on a table. Translating that "left" and "right" notion to two pieces of graph converging on a point made a HUGE difference. To be able to say a limit doesn't exist and recognize what that looks like is a big gain over previous years.