As has been noted above, this is the "alternate" definition of the derivative (or "derivative at a point")..
If f(t) = t^(1/3)
f'(t) is the above limit, with t in place of 2.
So the above limit is f'(2) = (1/3)*2^(1/3-1)
View the numerator as one factor in the difference of cubes, and multiply by ...(something I'd rather not type) to "rationalize".
Then you can use direct substitution.