Calculus of a Single Variable, Chapter 9, 9.1, Section 9.1, Problem 38

Before we start calculating the limit, we will simplify the expression for the general term of the sequence. To that end we shall use recursive definition of factorial.
n! ={(1 if n=0),(n(n-1)! if n>0):}
a_n=((n-2)!)/(n!) =((n-2)!)/(n(n-1)(n-2)!)=1/(n(n-1))
Now it becomes easy to calculate the limit and determine convergence of the sequence.
lim_(n to infty)a_n=lim_(n to infty)1/(n(n-1))=1/(infty cdot infty)=1/infty=0
As we can see the sequence is convergent and its limit is equal to zero.
The image below shows the first 15 terms of the sequence. We can see they are approaching x-axis i.e. the sequence converges to zero.