# Thread: Sign Functions and Limits Problem

1. ## Sign Functions and Limits Problem

The sign function sgn(x) is defined as follows:

sgn(x)={x/lxl if x does not equal 0; 0 if x=0}

Use the sign function to define two functions f and g whose limits as
x -> 0 does not exist, but such that

(a) lim [f(x)+g(x)] does exist
x->0
(b) lim (f(x))(g(x)) does exist
x->0

2. The easiest way of obtaining this would be for the one-sided limits to be different for both functions:

$\displaystyle f(x) = sgn(x)$
$\displaystyle g(x) = (-1) * sgn(x)$

Then:

$\displaystyle (f*g)(x) = (-1) * sgn^2(x) = -1$
$\displaystyle (f+g)(x) = (-1) * sgn(x) + sgn(x) = (-1+1)*sgn(x) = 0$

But,

$\displaystyle \lim_{x\to0^+} f(x) = \lim_{x\to0^+} sgn(x) = 1$

$\displaystyle \lim_{x\to0^-} f(x) =\lim_{x\to0^-} sgn(x) = -1$

Therefore $\displaystyle \lim_{x\to0^+} f(x) \neq \lim_{x\to0^-} f(x) \Rightarrow f(x)$ does not converge at $\displaystyle x=0$ and neither does $\displaystyle g(x)$ in the same way.

3. Sweet