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

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.

## Answers and Replies

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

arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
Let f(x) be any crazy function. Let g(x) = 1/f(x).

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. pasmith
Homework Helper
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.

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

HallsofIvy
Science Advisor
Homework Helper
Consider the function which is equal to 1 if its argument is rational and 0 otherwise.
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.