Calculus Limit Introduction

One problem I've had in years passed is kids getting a good grasp of what limits are and how to approach them. A few years back, I had a moment of clarity as a particularly confused set of seniors were staring back at me.  In that moment of frustration, I opened up Google and pulled up a map of the school.

I asked them this: If I had never been here before, how would you tell me to get to the school?  Kids jumped on this, giving me the address, the landmarks nearby and how to proceed from them, etc.  One kids went so far as to give me cardinal directions based on nearby freeways (and Houston has enough of those).

I then posed this: pretend that, unbeknownst to you, a bomb blew up underneath the school (please don't tell anyone I said that) and the whole thing flew up into the air and landed about 20 miles that way... were your directions wrong according to your best knowledge?

This sparks some debate.  Yes, because the school isn't there anymore.  No, because you didn't know it blew up.  Yes, because how could you not hear the explosion you live like a mile away.  In the end, they focus on the "to your best knowledge" part and say "No, the directions where right".  Good.

"Why?"

And how quickly they hit this phrase: "Because that's where the school was supposed to be"

From there, it's a quick jump to graphs and values and all that good stuff.

I've since refined that practice.  I start with that stuff now.  I give them a series of addresses/coordinates in the city and ask them to look them up and tell me what's there [f(c)] or what should be there[limit as x approaches c of f(x)].  Some are actually there (continuous) some are construction sites with a "coming soon" (it's a bit hard to find these on google maps unless you remember that something is a new building).  I'll also include two streets that don't connect properly.

"What's on the corner of Wilcrest and Barryknoll?"  "Which one?"  Bam, discussion of one-sided limits and limits failing to exist.   

Conceptually, this hits home pretty quickly.  Number crunching comes later, but by then it's easy since they just know what they're looking for.  The next step is to have them submit screenshots of places they find on maps and refer to them using limit notation.

Example:  Chuy's is located on 9350 Westheimer Rd.

Limit as x approaches 9350 of Westheimer equals Chuy's  (in actual limit notation, with arrow, etc.)

Westheimer(9350) = Chuy's 

Westheimer is continuous

Example:  They tore down a Starbucks on 1655 Voss Rd. Thanks, Obama! (not really, it's still there.)

Limit as x approaches 1655 of Voss Rd. equals Starbucks  (in actual limit notation, with arrow, etc.)

VossRd(1655) = Rubble 

VossRd has a removeable discontinuity at 1655

Thanks, Google.