Evaluate:       4 44 41 1lim 1 1nn nn n    

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

First, let's simplify the numerator: 4n^4 - 4n^2 + 1 = (2n^2 - 1)^2

Next, let's simplify the denominator: n^4 + n^2 = n^2(n^2 + 1)

Now, we can rewrite the expression as: lim (2n^2 - 1)^2 / (n^2(n^2 + 1))

To evaluate the limit as n approaches infinity, we can divide both the numerator and denominator by n^4, which is the highest power of n in the expression.

After dividing, we get: lim (2/n^2 - 1/n^4)^2 / (1 + 1/n^2)

As n approaches infinity, the terms with 1/n^4 and 1/n^2 become negligible compared to the other terms.

Therefore, the limit simplifies to: lim (2/n^2)^2 / 1

Simplifying further, we have: lim 4/n^4

As n approaches infinity, the value of 4/n^4 approaches 0.

Therefore, the final answer is: lim 4/n^4 = 0