You can put this solution on YOUR website! cube root of x is equal to x^(1/3)
fifth root of x is equal to x^(1/5)
the expression is equal to (x^(1/3))^(x^(1/5))
if x is 0, the expression becomes (0^(1/3))^(0^(1/5))
this becomes 0^0, since 0 to any power is equal to 0.
calculator says that can't be done because you can't have 0^0.
since anything raised to the 0 power is equal to 1, i would think that the limit as x approaches to 0 would be equal to 1.
i used excel to test that out.
my results indicate that what i suspected is true.
here's the results from excel.
i started with x = 1 and made it smaller by a factor of 1000 each time, i.e. 1 divided by 1000 = .001 = 1 * 10^-3.
that divided by 1000 becomes (1 * 10^-3)/(1*10^3) = 1*10^(-3-3) = 1*10^-6 = .000001.
the excel does that automaticlly by formula.
the progression goes on until the result is 1.
that's the value of the expression as x approaches 0, as best i can determine.
As x approaches 0 from the right, both and approach zero, but neither ever reaches 0.
Any non-zero number to the zero power is equal to 1. Since is not zero and is approaching zero, this limit is the limit of a non-zero number as the exponent approaches zero; that limit is 1.
