Instructor: Marcel Goh
Email address: marcel[dot]goh[snail]mail[dot]mcgill[dot]ca
Classes: Mondays, Wednesdays, and Fridays 11:35 to 12:35 in McIntyre 522.
Office hours: Wednesdays 13:30 to 15:30 in the grassy area near Milton Gates.
(I will be wearing a large black hat for visibility
purposes.) In case of inclement weather, I will hold office hours in McConnell
311 instead.
If this time slot does not work for you and you need to see me urgently,
you can email me to make an appointment.
Notes: A PDF of the course notes can be found here.
At any given time during the semester, they may
lag behind or be ahead of where we are in the course—I will update them from time to time
but not necessarily regularly. Sometimes I'll put things in the notes that I don't end up saying,
and sometimes I say things in class that are not in the notes. However, my intention is that at the
end of the semester,
these will be a somewhat faithful record of all that was covered during my lectures.
Please do not expect
these notes to be a substitute for attending class or for doing the exercises in the book, the latter
of which is still by far the best way you can prepare yourself for the exam.
Midterm exam: 19:00 to 21:00 on 29 October. Details about the location(s) will be announced
in due course, on myCourses.
Final exam: T.B.D., December.
31 August
Today we spent the first half of the class going over the syllabus and fretting about various
administrative
things that don't matter.
For the second half of the class, we pondered the idea of the "rate of change" of a function
"at" a given point.
We saw how taking successive approximations can get us closer and closer to this
"instantaneous rate of change."
2 September
We started today's class by thinking about what we would like our definition of a function being close
to something as its input approaches something to have. Then we stated this definition: that of a limit.
It was a long definition that is hard to digest, so we dwelt on the definition for quite some time.
Then we proved that the limit of a certain rational function equals 4 when its input is close to 1,
directly from the definition (without using any fancy laws or whatnot).
4 September
We continued thinking about the concept of limit this class. We stated formally what it means for a function
not to equal a given number, then interpreted the very definition of limit as the result of a game
between ourselves and an all-powerful Adversary. We also defined what it means for the limit to not
exist. Our running example throughout this entire class
was the function |x|/x, and we defined the concept of one-sided limits. Lastly, we learned
that if the one-sided limits exist but do not agree at a given point, then the limit does not exist there.
9 September
Today we started by establishing the fact that if the left- and right-hand limits of a function both exist
and are equal to one another, then the bona fide limit of that function equals that limit too. This
is a direct consequence of the definitions so we did not dwell on it too long. The bulk of this class was
spent defining vertical asymptotes and what it means for a function to go to infinity or minus infinity.
11 September
We discussed limits at infinity, including infinite limits at infinity. We considered a couple of examples,
then moved on to a new section on algebraic limit laws.
14 September
We began today with a proof sketch of the sum law for limits, then saw various examples of the limit laws
in action. The examples we discussed involved the use of the direct substitution rule, sometimes
by invoking a theorem that says that if two functions f and g agree in some neighbourhood
of a (except possibly at a itself), then their limits at a must be equal. We saw an example
of using one-sided limits to our advantage before noting a rule for limits at infinity that we'll use
next class.
16 September
We practised evaluating limits at infinity by trying our hand at a few examples, then introduced the squeeze
theorem and gave an application. This was the last lecture devoted solely to limits.