# 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

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.

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