Limits.sagews

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 $f(x) = 2x$, for example, the limit of $f(x)$ as $x$ approaches $4$ would simply be $8$, since $2 * 4$ is $8$. The notation for this, as you will surely see in a calculus book, in a calculus classroom or on a calculus test, looks like:

$\underset{x\to4}{\rm{lim}}2x=8$

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 $f(x) = \dfrac{x^2 - 1}{x - 1}$, as well as the code to generate such a graph.

# Create a plot of our function using variable 'x' on the domain -1<x<3
p = plot((x^2 - 1)/(x - 1), x, -1, 3)
# Create an open (faceted) point at (1,2) with a 'white' interior and the given point size
pt = point((1, 2), rgbcolor="white", pointsize=30, faceted=True)
# Combine the plot and the point (the plus sign indicates 'combine these elements')
g = p+pt
# Show the combined plot, using the given x and y bounds (similar to a graphing calculator)
g.show(xmin=0, xmax=3, ymin=0, ymax=4)

Looking at $f(x)$, one can see that setting $x$ equal to $1$ 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 $f(1)$ is undefined, however, we can still analyze, by way of limits, what $f(1)$ would equal if it did exist. The notation for this would be:

$\underset{x\to1}{\rm{lim}}x^2-\dfrac{1}{x}-1$

Just by looking at the graph, one can see that as $x$ approaches 1, the y-value for $f(x)$ approaches $2$. To find this value algebraically, we can remove the discontinuity by factoring the numerator, then dividing both the top and the bottom by $(x - 1)$ to obtain:

$\underset{x\to1}{\rm{lim}}x^2-\dfrac{1}{x}-1 = \underset{x\to1}{\rm{lim}}\dfrac{(x+1)(x-1)}{(x-1)} = \underset{x\to1}{\rm{lim}}x+1 = 2$

Thus, both graphically and analytically, we can see that the limit of $f(x)$ as $x$ approaches $1$ is equal to $2$. To verify this result, we can actually use the following code to have SageMathCloud compute the limit for us:

# Evaluate the limit of (x^2-1)/(x-1) as x approaches 1 (designated by 'x=1')
limit((x^2 - 1)/(x - 1), x=1)

2

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 $\pi$ from SageMath.

Practice Problems

Evaluate each limit graphically and analytically

1)$\underset{x \to 2}{\rm{lim}} f(x) = \dfrac{x^2-2x-8}{x-4}$

plot((x^2 - 2*x - 8)/(x - 4), x, -1, 5).show(xmin=0, ymin=0)

2)$\underset{x\to\pi}{\rm{lim}}\dfrac{1-\cos(x)^2}{\sin(x)}$

plot((1 - cos(x)^2)/sin(x), x, -1, 5).show(xmin=0, ymin=0)

3)$\underset{x\to2}{\rm{lim}}\dfrac{(x-2)^4}{x^2-4x+4}$

4)$\underset{x\to0}{\rm{lim}}\dfrac{\sin x}{x}$

5)$\underset{x\to0}{\rm{lim}}\dfrac{1-\cos x}{x}$

%md
<h4>To view answers, select this cell and paste it into your own sagews, and then double click on it</h4>
<!--
    1) Limit evaluates to 6
    2) Limit evaluates to 0
    3) Limit evaluates to 0
    4) Limit evaluates to 1 (graph it to see)
    5) Limit evaluates to 0 (graph it to see)
     -->

To view answers, select this cell and paste it into your own sagews, and then double click on it

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