<

The Concept of a Limit

---

The limit concept is the basis of calculus. Without it, there is no derivative, hence no calculus. Mathmaticians have refined this concept to a precise definition so as too confuse many non-math people and make it totally non understandable for many others. This formal definition is still taught today in a basic calculus I course. In my opinion, many get so confused by it, they lose motivation for the subject. As an RA during my undergraduate time, this was the #1 question students would come to my door with. That forced me to keep studying the concept and learn it well. Any teacher would tell you that you can not explain something to someone if you do not understand it yourself.

I will state the formal definition of a limit in the next article and show you a simple proof.

Basically, we want the f(x) values to get close to L when the x values get close to a on the x-axis. I think what confuses most students is the use of the absolute values along with the inequalities. Think of L as a level & (0,L) is on that level on the y-axis. Think of Î as a small positive number which measures the distance away from L both above & below L. This is call a neighborhood of L. For a given Î, we should be able to find a neighborhood of a on the x-axis (a positive distance d on both sides of a that will produce y values (f(x) values) that lie in the e neighborhood of L, no matter how small e becomes. If you can find a d (neighborhoods of a) for each Î (neighborhood of L), then the statement is justified (the limit is indeed L).

Don't worry about this, 85% of math majors don't really understand it either.

The only reason I'm spending time on it is because it's the basis of getting the derivative of a function. The limit process is used. Recall, we looked at the slopes of many secant lines on the graph as the moveable point approached the stationary point and used the limit in determining the limit of the slopes to be the slope of the tangent line to the curve at the stationary point. This is the derivative of the function at that point.

In a formal course in calculus much time is spent on limits and the analysis is very intensive. Many theorems are proven using that formal definition and rarely understood by many. However, the very best students have no problem with this and usually are very successful as math majors.