To provide the learner with a firm intuition regarding limits of sequences and limits of functions
To see several examples in which we reason with limits
This packet introduces limits of functions and limits of sequences, a fundamental concept in mathematical analysis.
This packet requires very little background. You should be familiar with functions and should be able to visualize some graphs of functions, but that's really about it.
The intuition behind the concept of a limit is this: as you move along a smooth line toward some point, you can keep getting closer and closer to the point, forever, without ever actually reaching it. Why? Because between any two points on the line, there are infinitely many points. Between 0 and 1, for example, are an infinity of intervening values. This notion, the notion of the continuum, is fundamental to understanding limits.
A limit is a bit like the paradox of running a foot race. Suppose I am running a foot race from A to B. Before I can get to B, I have to travel 1/2 of the distance, to a midpoint C. Upon reaching C, I have to travel half the distance from C to B, to some midpoint D. But in this case, once I reach D, I must travel half the remaining distance to a new point E. And this continues, forever, so that before I get to the destination, I must go half way there first - but I can keep going 1/2 the way for ever - therefore I can never finish the footrace. This is just like the idea of a limit - I get closer and closer to the destination B, and even though I never get there, after a while I'm so close that nobody can tell the difference anyway.
And with that in mind, we proceed to learn about limits.
This video initiates a discussion of limits using sequences, and continues on to consider limits of funcitons.
This video gives several examples of reasoning with function limits.