> I want to know what is the best way to explain the following > to one of my students. This is my first year teaching AP > Calculus AB. > > I gave the following two problems to my students for homework: > > 1. Find the limit of f(x) = sin (pi * x) as x approaches 1. > Some answered zero and the others answered DNE (does not exist) > > 2. Find the limit of f(x) = cos (1/x) as x approaches 0. > All answered DNE
> As a class discussion, we said that problem #2 doesn't have > a limit because the function is oscillating infinitely and > it never approaches a single height. One student said that > the y-values are bouncing between negative and positive values > as you approach zero from the left and right and the function > never settles on any one value. > > For problem #1, a student asked why does the limit exist for > this function because the sine graph is an oscillating function.
One possible method for #1 is substitution. Letting u = pi*x, we're looking at the limit as x approaches 1 of sin(u), which is equivalent to the limit as u approaches pi of sin(u). In the last equivalence, I'm making use of the fact that x --> 1 implies u --> pi, AND u --> pi implies x --> 1 (i.e. the operation "x --> 1" is equivalent to the operation "u --> pi"). Note that both implications are needed, which you can see by considering an example such as exp(-1/x^2) as x --> 0 replaced with, via u = x^2, exp(-1/u) as u --> 0. Of course, now students have to know that the limit of sin(u) as u approaches pi is equal to sin(pi) = 0, and I don't know how they're supposed to REALLY know this without essentially knowing that sin(x) is continuous at x = pi.
However, assuming this is mostly an informal approach to limits, here's one way to handle it. From a graphical standpoint, look at the behavior of the graph as you mentally trace along the graph towards the point with x-coordinate 1. Does the graph "come together" at a point as you trace towards the point from both sides? (Of course, it is irrelevant whether the "point of togetherness" is a point on the graph.) If YES, then the limit exists and the limit is equal to the y-coordinate of the "point of togetherness". If NO, then the limit doesn't exist. In this particular example, it might also be helpful to review the fact that the graph of y = sin(pi*x) is simply a horizontal stretch of the graph of y = sin(x). Yes, I know this won't be a review for some students and others will have totally forgotten it, so it provides an opportunity to squeeze in a few minutes of precalculus review in a setting relevant to them, so it will more likely stick with them.
In the case of #2, all the wiggles are squished into the vicinity of x = 0, so the graph doesn't "come together" there. Of course, you'll also want to consider the example y = x*sin(1/x) as x approaches 0, which also has infinitely many wiggles squished into the vicinity of x = 0, leading you to refine the explanation of y = sin(1/x) by saying that the wiggles don't damp out as you approach x = 0. Then there are the examples y = f(x) and y = x*f(x), where f is the function that equals 1 when x is rational and 0 when x is irrational, which don't really have wiggles but they seem similar to sin(1/x) and x*sin(1/x), leading to a more precise (but also more abstract) idea of "function fluxuations" damping out or not damping out as you trace towards a point. (And then maybe introduce the squeeze theorem for limits if this digression goes on long enough ...)