Chapter 1: Limits.

Section 1.2 - 1.3 (More)

 Limit notation and pictures:

(Here are a few notes from last year)

So, first, we need the new nomenclature of "the limit of f(x) as x approaches c (from the left, from the right, or just in general)," in order to describe what we see in pictures, then we need a formal definition to handle some of the strange situations that can arise.

Here's another example -- getting you used to the notation:

In the late 1700s, and the 1800s (especially), mathematicians became interested in some rather more bizarre functions, and in order to understand them, a more general, as well as more logically precise, definition of limit was needed.  Consider the following two very strange functions:

1)  f(x) = 1, if x is a rational number, and f(x) = 0, if x is an irrational number.  This is sometimes called the Characteristic (or Truth) Function for rational numbers.  So, f(0.5) = 1, but f(pi) = 0, for example.  The question is, even though you can't draw this function, can you think about limits?

2) g(x) = 0 if x is an irrational number, and g(x) = 1/q, if x is a rational number, p/q, written in lowest terms.  So, for example, g(0.75) = 1/4, and g(8.669) = 1/1000, while g(2) = 1.  Again, while it's impossible to draw this function, can you think about limits, as you get close to a particular value?

(We'll revisit these in section 1.5, and find out that the second example is a rather amazing function that is discontinuous at every rational number, but continuous at every irrational number....)

To formally investigate limits, we need the technical, "epsilon-delta" definition:

Here's an excellent link describing the epsilon-delta definition, along with several formal limit proofs:

http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html

And here's a nice Java applet describing the same thing:

http://www.scottsarra.org/applets/calculus/EpsilonDelta.html

And here's a "practical" example:

Here's an example of how one can find a limit directly from the definition, using epsilons and deltas:

The function is the famous function:  f(x) = xsin(1/x).  See the picture:

Here, I've scaled and translated the xsin(1/x) function.  See if you can find the relationship between delta and epsilon that proves that the limit as x approaches 1 of this function is 3 (as it appears in the picture.)

GeoGebra Applet:

Now, try the homework from section 1.2 (it's mostly pretty easy, even if this epsilon-delta stuff lost you.  There's only one question dealing with that....

Next (don't bail out, it gets waaay easier on the next page!!!)