For awhile we’ve been working on limits…and all associated whatnot. But today, we refined our definition and brought our understanding to the next level. We started class out by looking at some graphs that were discontinuous or continuous and discussed why it was either choice.

One of the new continuous graphs we learned about were graphs with cusps, i.e. where a graph suddenly changes direction while still being continuous. We related it to a hiker on a mountain. A man is hiking to the tip of the point, maybe hesitates, but then falls off the other side, onto the graph. There is no removable discontinuity or step discontinuity when a cusp is present.

We previously knew what a discontinuity was, but now we have defined continuity further. Two ideas we assume that must be present for a continuous graph are the limit of f(x) as x approaches c must exist and also that f(c) must exist. Basically, in a continuous graph limf(x) as x appoaches c = f(c).

Next, we talked about notation for “one-sided limits”. This is the same thing that we’ve previously been doing, just notated different. When we first learned limits, we did check both sides of the function. We’ll still use that notation, but we expanded it. For the limit that approaches point c on the negative, or left, side, we use the notation x?c-. For the limit that approaches point c on the positive, or right, side, we use the notation x?c+. If both of those limits match, then we can say that the limit as x?c is that number.

Which brings me to the next topic: using this notation, what is a limit?

A limit (L) equals the limit f(x) if and only if the limits from both the negative and positive branches match. If the limits of the branches do not match, then we say that the limit does not exist, abbreviated by dne.

Again, L=lim f(x) as x? c  iff  L=lim f(x) as x?c-  and  L=lim f(x) as x?c+ 

For the rest of class we worked on problems to apply our newly aquired knowledge. For more information on limits see:

http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx

http://www.mathnstuff.com/math/spoken/here/2class/420/limit.htm

For the record, the ? are right arrows that are used in limit notation. I’m not too sure why they don’t show up.