Prove this limit = 3/4.....?

Question

Show the steps to get the answer, 3/4:

lim (n -> ∞) [3^(n+1)]/[2^(2(n+1))] [2^(2n)]/[3^n] = 3/4

Explain your work. Thanks! :)

Answer 1

Actually all the n's cancel out in your expression.  I will re-write your expression as follows:

[3^(n+1) * 2^(2n)] / [2^(2(n+1)) * 3^n]

Notice that [3^(n+1)]/[3^n] = 3

Also [2^(2n)]/[2^(2(n+1))] = 1/(2^2) = 1/4

Thus your final answer = (3) * (1/4) = 3/4