We need to find the limit of the expression $\frac{n^2}{(n+1)^2}$ as $n$ approaches infinity.

1. Problem Description

We need to find the limit of the expression $\frac{n^2}{(n+1)^2}$ as $n$ approaches infinity.

2. Solution Steps

First, expand the denominator:

$(n+1)^2 = n^2 + 2n + 1$

Now, the expression becomes:

$\frac{n^2}{n^2 + 2n + 1}$

To find the limit as $n$ approaches infinity, we can divide both the numerator and the denominator by the highest power of $n$ in the denominator, which is $n^2$:

$\lim_{n \to \infty} \frac{n^2}{n^2 + 2n + 1} = \lim_{n \to \infty} \frac{\frac{n^2}{n^2}}{\frac{n^2}{n^2} + \frac{2n}{n^2} + \frac{1}{n^2}}$

Simplify the expression:

$\lim_{n \to \infty} \frac{1}{1 + \frac{2}{n} + \frac{1}{n^2}}$

As $n$ approaches infinity, $\frac{2}{n}$ and $\frac{1}{n^2}$ approach 0:

$\lim_{n \to \infty} \frac{2}{n} = 0$

$\lim_{n \to \infty} \frac{1}{n^2} = 0$

Therefore, the limit becomes:

$\lim_{n \to \infty} \frac{1}{1 + 0 + 0} = \frac{1}{1} = 1$

3. Final Answer

The limit is 1.