Hint:
3^n < {2^n+3^n} < 2*3^n
3^n < {2^n+3^n} < 2*3^n
==>
{3^n}^{1/n} < {2^n+3^n}^{1/n} < {2*3^n}^{1/n}
==>
{3^n}^{1/n} < {2^n+3^n}^{1/n} < {2}^{1/n} * {3^n}^{1/n}
==>
3 < {2^n+3^n}^{1/n} < {2}^{1/n} * 3
==>
lim{3} <lim[ {2^n+3^n}^{1/n}] < lim[{2}^{1/n} * 3 ]
==>
3 <lim[ {2^n+3^n}^{1/n}] < 3 lim[{2}^{1/n} ]
{ lim 2^1/n =1 }
==>
3 <lim[ {2^n+3^n}^{1/n}] < 3
Applying sandwich rule :
lim[ {2^n+3^n}^{1/n}] = 3
Let
Wow. And after all that I end up at Mr. Fantastics result. I'm actually depressed.
In any event, to evaluate Mr. Fantastics limit we will revert one step.
Note how the 1 over n will tend towards 0 as n goes to infinity as will what is in the ln portion of the equation. We will end up with resulting in the answer 3.