which of the following statements is true for all c and all L?

Question

I) lim f(x)= L ==> lim I f(x) I = I L I
x->c....................x->c

II) lim I f(x)= I L I ==> lim f(x)= L
x->c................... ....x->c

Answer 1

As the first poster mentioned, the second statement is false. In fact, there is a counterexample that fails worse than that. Let f(x) = {1 if x∈Q, -1 if x∉Q}. Then [x→0]lim |f(x)| = [x→0]lim 1 = 1, but [x→0]lim f(x) doesn't even exist, let alone equal 1.

The first one is true. Again, this follows from the continuity of the absolute value function.

Answer 2

Off the top of my head, I wouldn't have thought that either one was true. After thinking about it for a while, I can at least show that the second one is false.

Let f(x)=x, c=(-1), L=1. Then:

lim |f(x)| = 1 = |L|, however:
x-->c

lim f(x) = -1 = -L.
x-->c

So if one of the two statements must be true, it's the first one.