What's a limit?

This is the story behind a status I had on facebook (sort of sounds like the first line to a country song, doesn't it?)  So, my friend Brenna came over, with a perplexed look on her face, and said in her sad little freshman voice that she needed help with limits.  So, we sit down and I start to explain it, making about as much headway as a vegan in a cow eating contest.  My efforts, of course, were hampered by the peanut gallary (Tom, making my job harder, and Summer, who was just there to watch).  So, thinking quickly, I ask,
     "Alright, so what does a slope mean?"
     "Rise over run."  She said matter-of-factly (if a little confused).
     "But what does that mean?"  She thought for a moment, and then said,
     "How much you go up and then go over?"  Quickly I realized that this was a futile effort and that I would have to abandon it for another mode of thought.  So, I thought for a few moments and quickly hatched a brief thought of an idea.
     "Tom," I said, "stand against the wall."  He did as bidden; "Summer, stand next to him."  Which she did.  "Alright... um... which one of you is taller?"
     "I am."  Said Summer.  Then she looked at Tom, and then back at me.  True, she might have been taller by about three milimeters. Quickly thinking, Tom ducked down a little bit (I think to mess me up.)
     "Stay there!" I quipped.  Just then, Stina walked by.  I bade her to sit by the wall and next to Tom; she did.  Richard, hearing the commotion, pocked his head out of Tom's room.  "Lie down here."  I ordered and, to play along, he did.  I had a look at my handiwork and realized that I had created an almost perfect exponential function.  However, my work was far from over.  "Now," I said, returning my attention to my anxious pupil, "imagine that this is the curve in your problem [4x^2].  Now, the slope is the difference between the height of Richard and Stina," pointing them out as I went along, "Stina and Tom, Tom and Summer."  Boing; the lights went on behind Brenna's eyes.
    "I get it,"  She said, "I understand what a slope is."
    "Good,"I replied, angling my body to go across Tom's head, with my feet around Stina's "now, imagine that I [Austin] am a line tangent to Tom's head, since Tom is the point where x=4.  Find Austin"
     "Alright, how do I do that."  So, I had Tom stand up.
     "Alright, first what you have to do is find the slope of Austin.  Now, what is a slope?"
     "The difference between a point and it's closest neighbor."
     "Right, so, imagine that Tom is the point that you want, and Summer is his closest neighbor.  Now, if we find this slope, we'll be almost there.  However, we want to get a little closer, so we use an 'almost Summer', and then to get closer a 'not quite Summer', and then we start using, 'Almost Toms', getting progressively closer to Tom, without actually reaching his height.  Do you understand."
    "Yes."
    "Alright, so the formula that we use is:  (((Tom + next closest)-Tom)/(next closest)).  Does that make sense."
    "OMG Yes!  This is finally starting to make sense!"  And that is how I explained limits, and it was good.