Day 3 in Mat329

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Today:

Announcements
- Reminder: stow those rectangles....
- Reminder: quiz today, at the end of the hour.
- Reminder: homework due Monday.
- Whoops, I've forgotten to assign 14.2 problems; I'll do that soon.
- Roll
Last time we further discussed how to represent multivariate functions. We're like Plato, trying to deduce something of their world by looking at their shadows on a cave.
I encouraged you to think about using a coordinate system that one can actually see projected on a cave, rather than not:
Today we'll discuss limits, but from the multivariate perspective. First, however, let's take a look at one more set of exercises from 14.1 (I like this sort of exercise):
p. 915, 59-64
Can you determine which function is which, and what clues do you use?
Limits:
- As usual we start with the technical definition of a limit in the univariate case (found in section 1.7 of your book, p. 73). Let's take a look at that.
- We then ask "how must we modify this for multivariate functions?"
My son this morning asked "what are the ends of the past?" We talked about using time as an axis, and my son said that the past ends in two points. "Open or closed?" I asked. This is particularly important to the concept of limits and continuity.
"Open", he said, on the present (closed at the Big Bang). I asked him if there's an end to the past on "the present side", and he said "No, it just gets closer and closer to the present (but doesn't get to it)." The limit exists, and is the present. But the past never quite gets to the present.
In the univariate case, how can a limit fail to exist?
- Single holes are no problem;
- Jumps are bad (including infinite ones);
- Functions may fail to "settle down": sin(1/x) around x=0.
The same sorts of problems will occur for multivariate functions. But, ironically, limits may exist along particular lines of approach but not agree. This is something new and different. Let's take a look at p. 917, Tables 1 and 2.
Example 2, p. 919 is also interesting.
The generalization of the univariate limit of section 1.7 is on p. 917, and well illustrated on p. 918.
- The difference is in the domain, as usual. Instead of a segment surrounding a, we require a circle surrounding (a,b); in general, a ball surrounding the point in the domain at which we desire the limit, if it exists.
- Issued a challenge about getting within a tiny value $\epsilon$ (epsilon) of the limiting value L, we must be able to find a ball small enough (as indicated by radius $\delta$ ) that function values within that "delta-neighborhood" are within $\epsilon$ of L.

Next time: More 14.2: continuity

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