(A). The following function is an example of a "not nice" function. Using your graphing calculator (e.g., TI 83 or 84) or if you have access to some other graphing tool, graph the following function:
f(x) = sin (1/x) if x ≠ 0; = 0 if x = 0.
You are going to try and determine the value of limit of f(x) as x approaches 0.
If you are using a TI-83/84, start out by using the "standard" graphing window (ZStandard). What do you see?
Now change ONLY the y-window to YMIN = -1 to YMAX = 1. Does this help?
Now change ONLY the x-window to XMIN = -1 to XMAX = 1. What do you see now?
Keep reducing the size of the graphing window until any further reductions do not provide any more useful insight on the behavior of the function.
Think about the kind of analysis that you would need to perform to rigorously determine the limit of the function as x approaches 0? And what would this limit be?
(B). Consider the following function:
g(x) = x sin(1/x) if x ≠ 0;
= 0 if x = 0.
(i) The function g(x) is continuous at x = 0. Why?
(ii) Is the function g(x) differentiable at x = 0? Why? (Hint: Use your answer from part (A)).