## Wednesday, March 30, 2011

### Take it to the limit!

I skipped enough review stuff in Precalc this year that we're actually going to have a good chunk of time to introduce some limits.  I haven't gotten to this point for several years and am looking for ideas.

How do you introduce the idea of a limit?  I found Shawn's idea of using rolly chairs and stop watches to find speed and steer them towards the idea of a rate... which leads to slope... which leads to a limit.

I honestly don't know if I could pull that off (especially the rolly chairs thing).

Any other ideas?

1. i like to introduce limits as the number the *output value* gets close to. in other words, find the limit of the y-coordinate not the limit of the slope, which throws too many concepts together: (secant line, limit, tangent line, slope, derivative.)
you can then use the concept of limit to (later) make sense out of "the slope of a curve at a point."
for example, what's the limit of sequence 4/9, 49/90,499/900, ... ?

2. If you don't have to be formal and rigorous about it, you could just say "it's where the function approaches" -- and then give a few examples and make a worksheet which have students investigate limits for simple functions using tables&graphs on their graphing calculators (fxns like y=x^2, y=sin(x)/x, y=1/x as x->0 or limits of rational functions as x->infinity...).

But then you could also throw in a fxn like http://samjshah.com/2009/10/03/sin1x/ and have them investigate it graphically and numerically. i had my calc kids do this - we had groups and chart paper and each group had to present their observations, tables, conjectures, etc... eventually we proved that the limit does not exist, but you could show them the handwaving argument too...

or maybe you could find some other fun to investigate limits...

sam.

3. I introduced limits by assigning a modified version of Achilles & The Tortoise for homework and had students debate the paradox in class the next day.

Overall it was relatively successful, but there may or may not be a ton of scaffolding you need to get the students to actually debate.

4. Thanks for all of your help! I'm going to have to talk myself through them to figure out what works best for me. It'll probably be a combo of all three. :)