Help me to understand the solution..or give a new shorter solution

if we consider
a^n/n!
then with each increasing value of n
a^n/n!
decreases by factor of a/n+1 of the previus term
now for any value of a if n>4 the value of a^n/n! becomes less than 1
(u change check that by taking any arbitrary value of a)
so if n→infinity
a^n/n! is less than 0
and hence tends to 0 in the neighbourhood

I think thats what has been done in my solution...hasnt it ??

yes thats what has been done but in a other way......just to make it look difficult to students.........

if t_n→0
then t_n+1 must also →0
so can we have this t_n+1/ t_n ????
isnt it of 0/0 form ??

Or am I acting like a fooool :P

kkkk..

got it...

In general note that for large positive integer n,
log n < xⁿ < aⁿ < n!

That is the factorial increases faster than the exponential which in turn increases than the power function while the logarithm is the slowest among all.