some of my students have become paranoid.
while discussing multivariate limits, i distinctly remember telling them .. even writing an algοrithm/pseudocode for them:
- can you plug in? if there's no indetermιnate form, then all is well. [1]
- try simple trajectories first, like lines of varying slopes. if this gives two directional limits, then nothing more complicated is needed.
- otherwise, try something complicated. one of these things will work, but not both:
- try the squeezε theοrem, but make sure you actually have a correct inequality.
- try higher-order curves, like parabolas or cubics; the exponents are never larger than the exponents in the problem. [2]
others went, right away, to the squeezε theοrem trick, and writing out false inequalities.
in that class, i even spent time on one example showing a wrong inequality and why it's wrong ..
[sighs]
if they only stuck to the algo ..
[1] i've had very little luck explaining cοntinuity in a calculu∫ class. so when in rome, speak as the gladiatοrial crowds do.
[2] technically, it's not lying if i never give them a problem of that order .. \-:
4 comments:
I try to be even more concrete.
See the two terms in the denominator?
Set one of them equal to zero - that's a curve/line.
Set the other one equal to zero - another curve.
Set them equal to each other - the third curve.
If all results are the same, switch to proving that the limit exists.
give me a few more years of cynicism, L, and i might borrow your algo. q-:
"See that number to the right and above x? Write it down normal size. Put x next to it. Go back to that number and subtract 1 from it. Put the result to the upper right of x, in a smaller size."
The curve strategy is actually quite reasonable: either one of the terms dominates, or they are comparable - what else is there to look for?
I like Leo's method. I may have to try it. I usually have them substitute y=cx^n (n can be real) and then equate exponents with n's. But that still requires math but its relatively successful. (especially when the curve that fails is y=x^(1/3)
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