How do you calculate this limit: lim n→infinity ((n^2) *sqrt(3^n)) /(2^n)?

To calculate the limit of the expression as n approaches infinity:

lim (n → ∞) [(n²) * sqrt(3ⁿ)] / (2ⁿ)

You can use L’Hôpital’s Rule, which is a powerful technique for finding limits involving indeterminate forms like 0/0 or ∞/∞. Here’s how you can apply L’Hôpital’s Rule:

  1. Rewrite the expression to make it suitable for L’Hôpital’s Rule:

lim (n → ∞) [(n²) * sqrt(3ⁿ)] / (2ⁿ)

  1. Take the natural logarithm (ln) of both the numerator and denominator:

ln [lim (n → ∞) [(n²) * sqrt(3ⁿ)] / (2ⁿ)]

  1. Use properties of logarithms to simplify:

ln [lim (n → ∞) (n²) + ln(sqrt(3ⁿ)) – ln(2ⁿ)]

  1. Apply the power rule of logarithms:

lim (n → ∞) [2ln(n) + (1/2)ln(3) + nln(3) – nln(2)]

  1. Now, you can evaluate the limit term by term:

lim (n → ∞) 2ln(n) + (1/2)ln(3) + nln(3) – nln(2)

  1. As n approaches infinity, the natural logarithm of n also goes to infinity, so the first term becomes ∞.
  2. The other terms are constants, so they remain the same:

(1/2)ln(3) + ln(3) – ln(2)

  1. Combine the constants:

(1/2)ln(3) + ln(3) – ln(2) = ln(3^(1/2)) + ln(3) – ln(2)

  1. Use logarithm properties to combine the terms:

ln(√3 * 3) – ln(2)

  1. Simplify further:

ln(3√3) – ln(2)

  1. Subtract the logarithms:

ln(3√3 / 2)

So, the limit of the given expression as n approaches infinity is:

ln(3√3 / 2)

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