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One-Sided Limits and Continuity

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  • 0:06 Continuous and…
  • 1:22 One-Sided Limits
  • 2:41 Continuity
  • 3:34 Lesson Summary
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Lesson Transcript
Instructor: Robert Egan
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
Discontinuous Path Example

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.

One-Sided Limits

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
One Sided Limit Graph

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