Topics:
• Review of the concept of limits (for the benefit of several new students) and using tables in Desmos to guess the value of the limit (or whether or not there is a limit)
Here are links to the Desmos graphs and tables that I saved from class:
Showing that $\displaystyle \lim_{x\rightarrow 0}\frac{\sin(x)}{x} = 1$: notice that $\frac{\sin(x)}{x}$ is undefined when x=0, but the limit as x approaches 0 does exist and is equal to 1. The idea is that we are trying to guess what value the function “should” have when x=0, based on the values nearby.
Limit of a piecewise-defined function: This shows how to enter a piecewise-defined function in Desmos. Since the function is defined differently on each side of the target value of x, we have to use two tables, one for x approaching from the left and one for x approaching from the right.
Limit of a polynomial function: This is a table I made for problem 7, $\displaystyle \lim_{x\rightarrow 1}\left(x^2+3x-5)$ from the homework. Notice that in this case, the limit as x approaches 1 is -1, which is the same as f(1). This is a special situation!
It is very important that you understand what we are doing when we find a limit as x approaches some number c and how that is different from finding the value of the function when x=c. There are times when $\displaystyle \lim_{x\rightarrow c}f(x)$ will be equal to $f(c)$, but that does not mean that they are the same thing conceptually! In fact, that is a special situation that we will be discussing later when we get to the idea of continuity.
• Review of the three ways that a limit may fail to exist.
The limit fails to exist in these situations:
* when the function approaches different values from the left and from the right. (This tells us that in order for the limit to exist, there must be two “one-sided” limits and they must equal each other.)
* when the values of the function either grow without bound (“go to infinity”) or decrease without bound (“go to negative infinity”) as x approaches c. This happens when there is a vertical asymptote to the graph of the function. We will go into this in more detail later on.
* when the values of the function oscillate in such a way that they do not keep getting closer and closer to any fixed number, as x approaches c. (The example is $\sin\left(\frac{1}{x}\right)$)
• Average rate of change over an interval and instantaneous rate of change: I worked through my version of problem 1 from the WeBWorK except that I did not have time to complete the last part, where we are to find the limit of the $\frac{\Delta s}{\Delta t}$. If you have questions about how to compete this problem you can discuss on Piazza! See below.
Homework:
• If you have not already joined Piazza: Look for the invitation to join our Piazza discussion board, and join by following the link. Or you can go directly to Piazza here and join using your City Tech email address. We will be using the discussion board as a way to ask questions about the homework or the material discussed in class. (New students: I will send you your invitations sometime on Tuesday.)
• If you have not already done so, log in to WeBWorK following the instructions here (which were also handed out in class) and make sure to enter your City Tech email! (If you do not yet have a City Tech email, please enter it as soon as possible.) You must have your City Tech email address in your User information in WeBWorK in order to be able to use the “email your instructor” feature, and also it will be used to send out your midterm grades.
• Make sure that you have done all of the things that are in the First Day post and the post from last time!
• Do the WeBWorK assignment “LimitsIntroShort”. Make sure that you are using tables of values to find the limits in problems 9, 11, and 12! (If you go to a friend or tutor, they may try to show you some other way of finding these limits. Be insistent that you can only use the methods we have already learned in class!)
• There will be a Quiz on Wednesday. The question will be what was announced last time.
Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!
Note on my schedule: for the time being, I will only be able to be online to read and reply to emails at certain times of the day. (It is possible that I may be online at other times but I cannot guarantee it.) The times are roughly:
Monday – Friday early morning
Monday-Thursday around 2:30-3:00 PM
Sunday-Thursday evenings around 9-10 PM
Please be aware of this if you need to contact me by email. Thanks.
