Limits
Before we begin our discussion on limits, let's take a moment to become familiarized with the Sage Worksheet, which you will be using throughout these lessons. If you have not yet created an accoun on SageMathCloud, click on the "Account" button in the top right of your screen and follow the instructions. Once you have logged in, create a new project by clicking the "Projects" button in the top left corner of your screen, and press "New Project." After you have named your project, click on its name to view it. In the top left corner of your screen, press the "New" button, and create a file named "01-Limits.sagews." Once you have done this, you have successfuly created a Sage Worksheet, which you should feel free to edit. When there is a prompt in a tutorial to view given code in your own sagews, simply copy it from here and paste it into your own file, and evaluate it by pressing shift + enter.
A limit, to be concise, is the value that a function approaches as a variable (such as x) approaches a certain value. Most of the time, this is fairly straightforward. For a function , for example, the limit of as approaches would simply be , since is . The notation for this, as you will surely see in a calculus book, in a calculus classroom or on a calculus test, looks like:
Where limits will come in handy, though, is in situations where there is some ambiguity as to the value of a function at a point. As an example of this ambiguity, let's look at the graph of , as well as the code to generate such a graph.
Looking at , one can see that setting equal to would make both the numerator and the denominator equal to zero, which is why there is a circle at that point on the graph. Even though is undefined, however, we can still analyze, by way of limits, what would equal if it did exist. The notation for this would be:
Just by looking at the graph, one can see that as approaches 1, the y-value for approaches . To find this value algebraically, we can remove the discontinuity by factoring the numerator, then dividing both the top and the bottom by to obtain:
Thus, both graphically and analytically, we can see that the limit of as approaches is equal to . To verify this result, we can actually use the following code to have SageMathCloud compute the limit for us:
On your own, now, try evaluating the following three limits first graphically, then by algebraic simplification. To aid this pursuit, I have included SageMath code to plot the first two functions, though it won't circle the discontinuities for you this time. Simply copy the code for each into a new cell on your worksheet and evaluate it. For the last three, see if you can manipulate the code from one of the other examples to graph the function. On a sidenote, use "pi" without the quotes to reference from SageMath.
Practice Problems
Evaluate each limit graphically and analytically
1)
plot((x^2 - 2*x - 8)/(x - 4), x, -1, 5).show(xmin=0, ymin=0)
2)
plot((1 - cos(x)^2)/sin(x), x, -1, 5).show(xmin=0, ymin=0)
3)
4)
5)