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

[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.

 

[mathman]

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

 

[arildno]

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]

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]

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.