I was teaching algebraically finding limits today in precalculus. We got to the part where x --> infinity. I've tried various ways of explaining it in the past about how you only need to look at the highest power term on the numerator and denominator and make your decision that way.
I've likened it to "when you're plugging in REALLY LARGE NUMBERS, the largest power term is like the ocean and the smaller powered terms become like spit. They don't make much of a difference to the ocean volume." This stuck with some kids, but I don't know that all were convinced.
Well. I had one problem: (8x^3 + 2x - 7) / (-4x^3 + 5x + 100) or some such thing. On the graphing calculator, we entered 1,000,000,000 and stored it in x. Then we calculated 8x^3. Enter. Then we calculated 2x. Enter and compare. That semi-convinced them. THEN we calculated 8x^3 + 2x, and that really convinced them because it "was" the same output as 8x^3 by itself. And we all know that if the calculator says something, then it must be true.
If I was thinking (because what I did was on the fly and then I moved on), I should have also then (in the main window) plugged in the whole expression using the stored value of x .... or then putting in larger values of x and then "2nd entering" to recalculate the expression. Next time.