(1/2)* (1/3)^(1/0)=?

Others have pointed out that the expression you give is invalid but I understand that you are looking for the limit x->2 of the expression in your comment to Cpt M's answer.

1/(x-2) is approaching +∞ or -∞ depending from which side you approach it. This distinction is very important as you will see in a moment.

The important point is that this fraction is itself the exponent of ⅓.
Since the value of ⅓ is smaller than 1 it gets ever smaller, the more often you multiply it with itself:
⅓ = ⅓
⅓ * ⅓ = 1/9
⅓ * ⅓ * ⅓ = 1/27
⅓ * ⅓ * ⅓ * ⅓ = 1/81
and so on

If the exponent x of ⅓ is approaching infinity then (⅓)^x is approaching 0.
You could say that (⅓)^(+∞) = 0.

However, it's gets a bit more complicated.
Remember that the limit of your original expression is -∞ when you approach it from the left?
That means that the exponent of ⅓ is will approach -∞.
You may remember that a change of sign in the exponent of a fraction produces its multiplicative inverse.
(⅓)^(-1) = 3
Therefore (⅓)^(-∞) = 3^∞, which is also infinity!

So, the limit of your original expression will be 0 for lim[x->2+] and +∞ for lim[x->2-].

Hope this helps.

I think it's best to consider the limit of 1/n

as n ---> 0. Then it's clear that 1/n ---> ∞,

and thus (1/3)^(1/n) ---> 0.

1/0 doesn't equal anything, especially not infinity. If we approach 0 from the positive side, we get +infinity. If we approach 0 from the negative side, we get -infinity.