Math Does Not Equal Calculating

I have one question for anyone who has taken, or is taking calculus:  Can you explain what a Limit is?

I have asked this question literally dozens of times to engineering upperclassmen and graduate students.  Every single time its easy to see from their tortured attempts at an explanation that they have no real idea what a limit is in calculus.  They can integrate problems left and right, but they have virtually no idea why they are integrating something or what its purpose is.  

This is a fundamental failure of math education in this country.  We teach people how to calculate.  We don't teach people math.  

The fact is, calculations are a thing of the past.  Virtually all calculations performed in real world problems are automated with computers.  If memorization is the lowest form of learning, calculating is the lowest form of math.  It teaches nothing of importance other than how to follow a strict process to solve an over simplified problem no one will ever see in the real world.  And after 12-16 years of teaching math, no one has an idea WHY they are doing any of the calculations in the first place.

So I'll let you in on a little secret that most engineers eventually learn after college:  your math classes, and most of your math professors are a waste of time.  Most engineers don't really "understand" math until they start using it in the real world.  They quickly realize their schooling left them totally unprepared for what really matters:  learning how to correctly APPLY math (which is only a tool) to real world problems.

So let me answer that first question for you.  What is a limit?  Well, what is the smallest number of sides any shape can have?  One side only makes a line, not a shape.  Two sides will either make one or two distinct lines, again, not a shape.  A triangle, with three sides, is the shape with the least number of possible sides.  Four sides?  A rectangle.  eight sides?  An octagon...and so forth.

So what happens to a triangle, if we keep adding sides to it...over and over and over again.  Well, there is a limit that you will reach where your are no longer changing the shape of the object in any meaningful way.  Any idea what shape you will be left with?

A circle!  So, as you start to add sides to a triangle, from originally just 3 sides, to an infinite number of sides, the solution you will be left with is a almost perfect approximation of a circle. 

THATS IT!  That is the meaning and reason behind a limit.  There is no magic.  No black box.  Its simply a math trick that will allow you to come as close as possible (the limit) to the true shape of a circle, starting with something completely known...a triangle.

(Of course, limits can be applied to all kinds of things other than circles and triangles...things that exist in real life...like imperfect curves or surfaces...)