Last time, we reviewed a bit of trigonometry. Please read through appendix A in your textbook or consult the review posted on Courseweb. Limits All of calculus relies on the notion of a limit. You should think of a limit as the ultimate result of some trend or behavior. For example, consider two points with a distance of 1 between them and suppose you want to move from one point to the other. You decide to move in the following way: 1. You move half of the distance: 1 / 2 unit. Remaining distance is 1 / 2. 2. You then move half of the remaining distance: 1 / 4 unit. Remaining distance is 1 / 4. 3. You then move half of the remaining distance: 1 / 8 unit. Remaining distance is 1 / 8. 4. You repeat this process indefinitely. An illustration of the remaining distances is shown below with the starting point on the right hand side and the final point on the left hand side: The question you should think about very carefully is: do you ever reach the other point? Obviously, you get closer and closer to the other point since the remaining distance is shrinking. But, there is always a remaining distance. After 10,000,000 steps the remaining distance is 1 / 2 10 , 000 , 000 . Yes, this is small, but it is still nonzero so you have technically not traveled to the other point. No matter how many steps we take, if we travel in this way we will never reach the other point. But! We get arbitrarily close to the final point. We can make the remaining distance as small as we want just by taking more and more steps towards the final point. So, if we somehow imagine that we are able to take all possible steps, that is infinitely many steps, then we will arrive at the other point because the distance will have shrunk to 0. This is the idea behind a limit: it is expected result of a trend or pattern, even if that result may never occur. In our example, we would say that the limit of our behavior is that we reach
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- Spring '12
- Notes, Continuous function, One-sided limit