Over the river and through the woods is only fun on a continuous path. What happens when the path has a discontinuity? In this lesson, learn about the relationship between continuity and limits as we walk up and down this wildlife path.
Continuous and Discontinuous Paths
Following the earthquake, the path becomes discontinuous
Consider for a minute a sidewalk that goes over the river and through the woods and perhaps over some hills. If I take a look at the elevation of the sidewalk as a function of location, then its function might look something like this. Now this is a continuous path; I can trace this path without lifting up my finger, so what about the limits along this continuous path? Well, If I look at the elevation as I approach the treeline, I might find that the elevation is 100 feet. Let's say there was a gigantic earthquake! And the earthquake split the ground at the treeline. Now, if I approach the treeline from the river, then the limit might be 100 feet. But if I approach the treeline from the woods, then the limit might be 120 feet. If I want to trace this path, it's now discontinuous; I have to lift my finger up from the paper to continue tracing it because of this discontinuity at the treeline.
What can we learn from our treeline? First, limits can be different when you approach a point from the left- or right-hand side. These are called one-sided limits. A mathematical example of this might be the function f(x) where it equals x for x<1 and it equals x + 1 for x is greater than or equal to 1. This is a lot like our earthquake example. For values less than 1, f(x)=x. At 1, this line jumps because f(x)=x + 1. At this point here, we have a limit approaching 1 on the left-hand side that's different from the limit approaching 1 from the right-hand side. So let's look at the limit from the left-hand side. We're going to differentiate this limit from the limit that's approaching 1 from the right-hand side by putting a minus sign by the number that we're approaching. The limit as x approaches 1 from the left side is 1, and the limit as x approaches 1 from the right side - which is designated by a plus sign - is 2.
Example of a one-sided limit
The second thing we may have learned from our earthquake example is a little less obvious. Before the earthquake, the path was continuous, and before the earthquake, the limit as x approached some number, let's call it C, was independent of which side you took the limit. So you could approach the treeline from the left-hand side and get to 100 feet, and you could approach the treeline from the right-hand side to get to 100 feet. This was true across the whole path. After the earthquake, we had a discontinuous path. In particular, the limit of the elevation as we approached the treeline was undefined. Instead, we had to approach the treeline from either the river side or the tree side, and those two limits were different by about 20 feet.
From this, we learned a very important thing about continuity. We learned that a function, like y=f(x) is continuous in a region if the limit of that function as you approach any number equals the value of the function at that number. What this really means is that if you're approaching any point along your path, you can approach it from any direction and get to that point. There's not a discontinuity at that point either. Our path was continuous before the earthquake, and limits behaved nicely everywhere. You weren't going to all of a sudden fall off the face of the Earth as you were walking along the path.
After the earthquake, our path was discontinuous, and limits didn't behave nicely everywhere. For instance, if you were walking from the trees to the river, then all of a sudden you were going to fall off the path at the treeline because your path had a discontinuity.