# Homework Help: Show by means of example that lim/x→a/[f(x)g(x)] may exist

1. Sep 18, 2013

### MathewsMD

Show by means of example that lim/x→a/[f(x)g(x)] may exist even though neither lim/x→a/f(x) nor lim/x→a/g(x) exists.

I have tried using examples such as piecewise functions and rational functions, but can never validate the statement.

Any guidance and help would be great.

Thanks.

2. Sep 18, 2013

### mathman

Let f(x) be any crazy function. Let g(x) = 1/f(x).

3. Sep 18, 2013

### arildno

You forgot to insert Arildno's corollary:
"Let f(x) be any crazy function. Let g(x) = 1/f(x). THEN, g(x) is most likely also a crazy function"

Not very useful in this context, of course, but the result is beautiful, nonetheless.

4. Sep 18, 2013

### pasmith

Consider the function which is equal to 1 if its argument is rational and 0 otherwise.

5. Sep 18, 2013

### HallsofIvy

I presume you mean "let f(x)= 1 if x is rational, 0 if x is irrational.

And then let g(x)= 0 if x is rational, 1 if x is irrational.

fg(x)= 0 for all x so it trivially differentiable.