lim3^nn^3 -lim3^nn!

We can simplify the expressions using the ratio test.

• lim 3^n/n^3: divide by (3/2)^n
  • lim (3/2)^n / n^3
  • Using L'Hopital's rule, we get lim (ln(3/2)/3) / n^3 which is 0 as n goes to infinity.
  • Therefore, lim 3^n/n^3 is 0.
• lim 3^n/n!: divide by (3/2)^n
  • lim (3/2)^n / n!
  • Using Stirling's approximation, we get lim (3/2)^n / sqrt(2πn) (n/e)^n which is infinity as n goes to infinity.
  • Therefore, lim 3^n/n! is infinity.

In summary:

• lim 3^n/n^3 = 0
• lim 3^n/n! = infinit