1_10_2019 (1).pdf - Last time we reviewed a bit of...

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