limit n---infinity(a^n)/n!

Look at this problem in this way:

(a^n)/n! = (a/1)(a/2)........(a/a)......(a/k)....

these all the terms are in multiplication.

So, in denominator, it'll start from 1 and in b/w it'll be the time when denominator will be equal to a and then terms in deno starts increasing.

The number in denominator will approach to infinity.

and (a/n) where n is close to infinity is Zero.

So, the whole term multiplied by 0 becomes 0.

Hence, the ans is ZERO.

 let

y = (a^n)/n!

Take log both sides

i.e.

log y = log{(a^n)/n!}

log y = log {(a^n} - log n!

=> n log a - log {n(n-1)(n-2)....................1}

=>1/n [ log (a/n) +log a/n-1 + ............+ log a/1]

using limit to sum 

Log y = definite integral from 0 to 1 of log ax 

log y = log (ae^a)^1/a

=> y = a^1/ae