Evaluate:       2 ! 1 !lim 2 ! 1 !nn nn n   

To evaluate the given limit, we can start by simplifying the expression inside the limit.

Let's break down the expression step by step:

1. First, let's simplify the numerator:

   • The term (2n + 1) can be expanded as 2n + 1 = 2n + 1.
   • The term (2n - 1) can be expanded as 2n - 1 = 2n - 1.

   So, the numerator becomes: (2n + 1)(2n - 1).

2. Next, let's simplify the denominator:

   • The term (n + 1) can be expanded as n + 1 = n + 1.
   • The term (n - 1) can be expanded as n - 1 = n - 1.

   So, the denominator becomes: (n + 1)(n - 1).

3. Now, let's simplify the entire expression by dividing the numerator by the denominator:

   • (2n + 1)(2n - 1) / (n + 1)(n - 1) = (4n^2 - 1) / (n^2 - 1).

4. Finally, let's evaluate the limit as n approaches infinity:

   • As n approaches infinity, the terms with lower powers become negligible compared to the terms with higher powers.
   • So, we can ignore the constant terms (-1 and -1) in the numerator and denominator.

   Therefore, the simplified expression becomes: 4n^2 / n^2.

5. Now, we can evaluate the limit:

   • As n approaches infinity, the ratio of the highest power terms in the numerator and denominator determines the limit.
   • In this case, the highest power term is n^2 in both the numerator and denominator.

   So, the limit is: lim(n->∞) 4n^2 / n^2 = lim(n->∞) 4 = 4.

Therefore, the given limit evaluates to 4.