Perhaps someone can help me with a question that came up in class this morning. My 12th grade students were working on an assignment on limits. Today was the first class after spring break, and I had begun class by reviewing what they were supposed to have learned about limits already. I particularly emphasized the different techniques which they could use to find the values of limits that initially evaluate to 0/0. (They have not learned L'Hospital's rule, and they only know how to take derivatives of polynomials, since it is not primarily a calculus course.) We discussed factoring, graphing, and tables, with the graphing and tables done on our TI-83s.
One of the problems was the limit as x -> 0 of (1 - cos x)/x^2. I "know" that the answer is 1/2:
My students can prove it using trigonometry and the theorem about the limit of (sin x)/x as x -> 0. Multiplying by 1 + cos x and using a Pythagorean identity gives two factors of (sin x)/x and one of 1/(1+ cos x).]
I can also prove it (to myself, if not to my students) by using L'Hospital's rule. I have to differentiate twice.
And my students can figure it out by graphing the function in a reasonable window or making a table of values...or so I thought.
One of my students called me over because she couldn't figure out if the limit was equal to .5 or 0. She had chosen to make a table on the calculator. She had entered the x-values herself, and this is what she saw. x y1 .1 .49958 .01 .5 .001 .5 1E-4 .5 1E-5 .5 1E-6 0 1E-7 0 1E-8 0 Since I "knew" the right answer, I told her that I agreed with the .5 answer, and pressed the graph button to show her a picture which also suggested that the answer was 1/2. But the failure of the calculator to confirm this bothered me, and later I fired up a graphing program which can handle a higher degree of precision on my computer.
To my surprise, I got essentially the same result: A table of values showed a sudden drop to zero on the vicinity of 0, and a zoomed-in graph showed a wildly fluctuating graph on the interval [-0.0001, 0.0001]. The fluctuations were strange, but not impossible to understand. I figured it might settle down if I zoomed in still farther. But what I discovered was that on a interval of roughly [-1E-8,1E-8], the graph dropped abruptly and sat precisely on zero (to 10 significant digits). Of course, it's undefined at zero, but the graphical limit was _not_ the expected 1/2.
I consulted with another teacher who verified my calculated result of 1/2, but neither of us can explain why the technology suggests something completely different. My friend thinks part of the problem is the "connecting the dots" that the calculator does, but when we graphed it on the calculator in the dot mode, we still got the drop to zero.
Can anyone provide an explanation or insight? I plan to show my student the interesting graph I got on the computer when I see her again Thursday, but I'd like to be able to explain the discrepancy as well.