Limit of a limit?

Let X= $\large \lim_{k\to 1} \lim_{a\to 1} \lim_{n\to \infty} \left(\prod_{j=1}^{k}\left ( a^{-nk}(a+ \frac{j}{n})^{n}\right) \right)$

= $\large \lim_{k\to 1} \lim_{a\to 1} \lim_{n\to \infty} \left(a^{-nk}[(a+ \frac{1}{n})(a+ \frac{2}{n})....(a+ \frac{k}{n})]^{n}\right)$ = $\large \lim_{k\to 1} \lim_{a\to 1} \lim_{n\to \infty} \left([(1+ \frac{1}{an})^{n}(1+ \frac{2}{an})^{n}....(1 + \frac{k}{an})^{n}\right)$ [Introducing each 'a' into all the k brackets]

Now, Taking Log of both sides,

logX = $\large \lim_{k\to 1} \lim_{a\to 1} \lim_{n\to \infty} \left(n[log(1+ \frac{1}{an})+log(1+ \frac{2}{an})....(1+ \frac{k}{an})]\right)$

Changing n= $\frac{1}{s}$ , we get

= $\large \lim_{k\to 1} \lim_{a\to 1} \lim_{s\to 0} \left(\frac{[log(1+ \frac{s}{a})+log(1+ \frac{2s}{a})....(1+ \frac{ks}{a})]}{s}\right)$ = $\large \lim_{k\to 1} \lim_{a\to 1} \lim_{s\to 0} \left(\frac{1}{a}\frac{[log(1+ \frac{s}{a})}{\frac{s}{a}}+\frac{2}{a}\frac{[log(1+ \frac{2s}{a})}{\frac{2s}{a}}....\frac{ks}{a}\frac{[log(1+ \frac{ks}{a})}{\frac{ks}{a}}]\right)$
Since, $\lim_{s\to 0} \left(\frac{log(1+s)}{s} = 1\right)$

= $\large \lim_{k\to 1} \lim_{a\to 1} \left(\frac{1}{a}+\frac{2}{a}+......\frac{k}{a})\right)$ Therefore, X = $\large \lim_{k\to 1} \lim_{a\to 1} \left(e^{\frac{k(k+1)}{2a}}\right)$ = e