途中で定積分に持ち込みます。

x = n/(n!)^(\(\frac{1}{n}\)) と置くと、

log x = log n - \(\frac{1}{n}\)・∑[k=1,n] log k
log x = \(\frac{1}{n}\)・∑[k=1,n] log n - \(\frac{1}{n}\)・∑[k=1,n] log k
log x = -\(\frac{1}{n}\)・∑[k=1,n] log(\(\frac{k}{n}\))

ここで、
lim -\(\frac{1}{n}\)・∑[k=1,n] log(\(\frac{k}{n}\))
= -∫[0,1] log t dt
= [ t - tlogt ][0,1]
= 1　(∵lim[t→+∞] tlogt = 0)

lim[n→∞] log x = 1 より、lim[n→∞] x = e