Calculus of a Single Variable, Chapter 9, 9.1, Section 9.1, Problem 21

a_n = (5n^2)/(n^2+2)
To determine the limit of this sequence, let n approach infinity.
lim_(n->oo) a_n
=lim _(n->oo) (5n^2)/(n^2+2)
To solve, factor out the n^2 in the denominator.
=lim_(n->oo) (5n^2)/(n^2(1+2/n^2))
Cancel the common factor.
= lim_(n->oo) 5/(1+2/n^2)
Then, apply the rule lim_(x->c) (f(x))/(g(x)) = (lim_(x->c) f(x))/(lim_(x->c) g(x)) .
= (lim_(n->oo)5)/(lim_(n->oo) (1+2/n^2))
Take note that the limit of a constant is equal to itself lim_(x->c) a = a.
Also, if the rational function has a form a/x^m , where m represents any positive integer, its limit as x approaches infinity is zero lim_(x->oo) (a/x^m) = 0 .
(lim_(n->oo)5)/(lim_(n->oo) (1+2/n^2))
= 5/1
=5
Therefore, the limit of the given sequence is 5.