Use the theorem about sequences, i.e. direct substitution.

2(2)-3(3)=4-9=-5

Since lim bn does not equal to zero. We can use division rule.2. Assume that an-->2 and bn-->3 Evaluate lim x->inf (an-2)/bn

(2-2)/3=0

My favorite, the "squeeze theorem".3.We know from trignometrythat |sin(n)|<=1 for all n.This shows that 0<= |sin(n)/n|<= 1/n and hence that -1/n <=|sin(n)/n| <= 1/n. We can also evaluate the lim x->inf (sin(n)/n by using

(Other favorite is composite function rule).

We have that,(4) find lim x-->inf sin(n)/n

|sin(n)/n|<1/n

Thus,

-1/n<sin(n)/n<1/n

Squeeze theorem.

Both -1/n and 1/n ---> 0

Thus, limit is zero.