Find the lim x->infinite (x/[3x+5]) where [] denotes the greatest integer function use the squeeze theorem.

Answer 1

The greatest integer function of x is the equivalent of the floor function, and is the greatest integer that does not exceed x.

Since [x+3] is the floor function of x+3,
we have x+2≤[x+3]≤x+3

Let
L1=Lim x→∞ x/(x+2), and
L2=Lim x→∞ x/(x+3)

Evaluating, L1=1, L2=1, and therefore by the squeeze theorem (sandwich theorem),
lim x→∞ x/[x+3] = 1