↑ stuart clark:
Basically, the problem can be solved easily...
First, we solve an auxiliary corresponding limit
The value L
can be calculated by straightforward utilization of the Stolz-Cesàro theorem:
Now, using the Stirling approximation of the factorial
one gets for the value L
Finally, the continuity of the exponential function implies that the limit in the OP is equal to exp (L
denotes the Euler number.